On Matrices of Generalized Octonions (Cayley Numbers)

Abstract

This article focuses on generalized octonions which include real octonions, split octonions, semi octonions, split semi octonions, quasi octonions, split quasi octonions and para octonions in special cases. We make a classification according to the inner product and vector parts and give the polar forms for lightlike generalized octonions. Furthermore, the matrix representations of the generalized octonions are given and some properties of these representations are achieved. Also, powers and roots of the matrix representations are presented. All calculations in the article are achieved by using MATLAB R2023a and these codes are presented with an illustrative example.

1. Introduction

After Sir Rowan Hamilton discovered quaternions, Graves defined a new number system in 1843, which he named octonions [1]. Despite writing to Hamilton about his discovery in December 1843, Cayley independently arrived at the same number system, known as the Cayley numbers or Cayley algebra, and published his complete results in an article [2]. Afterwards, Hamilton explained that Graves had found and named these numbers before Cayley. This is why they are sometimes referred to as octonions and sometimes as Cayley numbers in the literature.

Octonions are primarily applied in theoretical physics. In the 1970’s, researchers attempted to use octonionic Hilbert spaces to study quarks. These structures are relevant to spacetime dimensions where super-symmetric quantum field theories are feasible. There have also been attempts to construct the Standard Model of elementary particle physics from octonionic frameworks [3].

Research involving octonions has extended to black hole entropy, quantum information science, string theory, and image processing. They have also proven useful in addressing the hand-eye calibration problem in robotics [3]. The Cayley algebra, which is non-associative and based on the Cayley numbers, is constructed according to the multiplication rules outlined in

Table 1. The multiplication table of Cayley numbers.

×	$u_1$	$u_2$	$u_3$	$u_4$	$u_5$	$u_6$	$u_7$	$u_8$
$u_1$	$u_1$	$u_2$	$u_3$	$u_4$	$u_5$	$u_6$	$u_7$	$u_8$
$u_2$	$u_2$	${\mu }_1{u}_1$	$-u_4$	$-{\mu }_1{u}_3$	$-u_6$	$-{\mu }_1{u}_5$	$u_8$	${\mu }_1{u}_7$
$u_3$	$u_3$	$u_4$	${\mu }_2{u}_1$	${\mu }_2{u}_2$	$-u_7$	$-u_8$	$-{\mu }_2{u}_5$	$-{\mu }_2{u}_6$
$u_4$	$u_4$	${\mu }_1{u}_3$	$-{\mu }_2{u}_2$	$-{\mu }_1{\mu }_2{u}_1$	$-u_8$	$-{\mu }_1{u}_7$	${\mu }_2{u}_6$	${\mu }_1{\mu }_2{u}_5$
$u_5$	$u_5$	$u_6$	$u_7$	$u_8$	${\mu }_3{u}_1$	${\mu }_3{u}_2$	${\mu }_3{u}_3$	${\mu }_3{u}_4$
$u_6$	$u_6$	${\mu }_1{u}_5$	$u_8$	${\mu }_1{u}_7$	$-{\mu }_3{u}_2$	$-{\mu }_1{\mu }_3{u}_1$	$-{\mu }_3{u}_4$	$-{\mu }_1{\mu }_3{u}_3$
$u_7$	$u_7$	$-u_8$	${\mu }_2{u}_5$	$-{\mu }_2{u}_6$	$-{\mu }_3{u}_3$	${\mu }_3{u}_4$	$-{\mu }_2{\mu }_3{u}_1$	${\mu }_2{\mu }_3{u}_2$
$u_8$	$u_8$	$-{\mu }_1{u}_7$	${\mu }_2{u}_6$	$-{\mu }_1{\mu }_2{u}_5$	$-{\mu }_3{u}_4$	${\mu }_1{\mu }_3{u}_3$	$-{\mu }_2{\mu }_3{u}_2$	${\mu }_1{\mu }_2{\mu }_3{u}_1$

[4].

The Cayley numbers, denoted by $\mathfrak{C}$, refer to the set that forms the Cayley algebra, commonly known as the octonions.

$$\mathfrak{C}=\{c=a_1{u}_1+a_2{u}_2+a_3{u}_3+a_4{u}_4+a_5{u}_5+a_6{u}_6+a_7{u}_7+a_8{u}_8|a_i\in \mathbb{R}\}$$

It is the fact that $\mathfrak{C}$ includes some number sets for special cases.

• When ${\mu }_1={\mu }_2={\mu }_3=-1$, this condition yields the real octonions [5].
• When ${\mu }_1={\mu }_2=-1,{\mu }_3=1$, this condition yields the split octonions [6].
• When ${\mu }_1={\mu }_2=-1,{\mu }_3=0$, this condition yields the semi octonions [7].
• When ${\mu }_1={\mu }_2=1,{\mu }_3=0$, this condition yields the split semi octonions [8].
• When ${\mu }_1=-1,{\mu }_2={\mu }_3=0$, this condition yields the quasi octonions [9].
• When ${\mu }_1=1,{\mu }_2={\mu }_3=0$, this condition yields the split quasi octonions [10].
• When ${\mu }_1={\mu }_2={\mu }_3=0$, this condition yields the para octonions [11].

There are a lot of articles about these octonionic type numbers such as [12,13,14,15,16]. In [17], the physical signal s can be represented by an 8-dimensional number, which is an element of octonionic algebra, given by: $s=ct+x_n{J}^n+h{\lambda }_n{j}^n+chwI,$ where $n=1,2,3.$ In this context, time is denoted by t, special coordinates are represented by $x^n$, and ${\lambda }^n$ refers to certain quantities. The dimensions of ${\displaystyle \frac{1}{\mathrm{momentum}}}$ and w are those of ${\displaystyle \frac{1}{\mathrm{energy}}}$. Additionally, the authors provided values for Planck’s constant and the speed of light. In their study, the authors linked the eight real parameters of octonions to spacetime coordinates, momentum, and energy, forming a complete octonionic basis. This description requires only the multiplication and distribution rules for three vector-like elements, facilitating the visualization of space’s 3-dimensional structure.

In literature, Cayley numbers is called also generalized octonion [18,19]. Furthermore, when viewed from the same perspective in the construction of 3-parameter generalized quaternions known in the literature [20], Cayley numbers is also be called 3-parameter generalized octonions (3PGO).

Any Generalized Octonion c is written in the form $c=a_1{u}_1+{\sum }_{i=2}^8{a}_i{u}_i=S_c+V_c,$ where $S_c$ is called the real part and $V_c$ is called the vector part of c. The summation of the Generalized Octonion $c_1=a_1{u}_1+a_2{u}_2+a_3{u}_3+a_4{u}_4+a_5{u}_5+a_6{u}_6+a_7{u}_7+a_8{u}_8$ and $c_2=b_1{u}_1+b_2{u}_2+b_3{u}_3+b_4{u}_4+b_5{u}_5+b_6{u}_6+b_7{u}_7+b_8{u}_8$ is

$$c_1+c_2=(a_1+b_1)u_1+\sum _{i=2}^8(a_i+b_i)u_i=S_{c_1+c_2}+V_{c_1+c_2}.$$

Also, these two Generalized Octonion are called equal when $a_i=b_i$ for $i=1,2,...,8$. Multiplication a Generalized Octonion $c=a_1{u}_1+a_2{u}_2+a_3{u}_3+a_4{u}_4+a_5{u}_5+a_6{u}_6+a_7{u}_7+a_8{u}_8$ with a real scalar k is defined as

$$kc=k{a}_1{u}_1+k{a}_2{u}_2+k{a}_3{u}_3+k{a}_4{u}_4+k{a}_5{u}_5+k{a}_6{u}_6+k{a}_7{u}_7+k{a}_8{u}_8$$

which satisfies the following properties:

$$\begin{array}{ccc}& & k(c_1+c_2)=k{c}_1+k{c}_2\hfill \\ & & (k_1+k_2)c=k_{1}c+k_{2}c\hfill \\ & & 1c=c.\hfill \end{array}$$

By taking $u_1=1$, $u_2=e_1$, $u_3=e_2,...,$$u_8=e_7$, the Generalized Octonion product of $c_1=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7$ and $c_2=b_0+b_1{e}_1+b_2{e}_2+b_3{e}_3+b_4{e}_4+b_5{e}_5+b_6{e}_6+b_7{e}_7$ is defined as follows:

$$\begin{array}{ccc}\hfill c_1\times c_2& =& a_0{b}_0+{\mu }_1{a}_1{b}_1+{\mu }_2{a}_2{b}_2-{\mu }_1{\mu }_2{a}_3{b}_3+{\mu }_3{a}_4{b}_4-{\mu }_1{\mu }_3{a}_5{b}_5-{\mu }_2{\mu }_3{a}_6{b}_6+{\mu }_1{\mu }_2{\mu }_3{a}_7{b}_7\hfill \\ & +& (a_0{b}_1+a_1{b}_0-{\mu }_2{a}_2{b}_3+{\mu }_2{a}_3{b}_2-{\mu }_3{a}_4{b}_5+{\mu }_3{a}_5{b}_4-{\mu }_2{\mu }_3{a}_6{b}_7+{\mu }_2{\mu }_3{a}_7{b}_6)e_1\hfill \\ & +& (a_0{b}_2+a_2{b}_0+{\mu }_1{a}_1{b}_3-{\mu }_1{a}_3{b}_1-{\mu }_3{a}_4{b}_6+{\mu }_1{\mu }_3{a}_5{b}_7+{\mu }_3{a}_6{b}_4-{\mu }_1{\mu }_3{a}_7{b}_5)e_2\hfill \\ & +& (a_0{b}_3+a_3{b}_0+a_1{b}_2-a_2{b}_1-{\mu }_3{a}_4{b}_7+{\mu }_3{a}_5{b}_6-{\mu }_3{a}_6{b}_5+{\mu }_3{a}_7{b}_4)e_3\hfill \\ & +& (a_0{b}_4+a_4{b}_0+{\mu }_1{a}_1{b}_5+{\mu }_2{a}_2{b}_6-{\mu }_1{\mu }_2{a}_3{b}_7-{\mu }_1{a}_5{b}_1-{\mu }_2{a}_6{b}_2+{\mu }_1{\mu }_2{a}_7{b}_3)e_4\hfill \\ & +& (a_0{b}_5+a_5{b}_0+a_1{b}_4-a_4{b}_1+{\mu }_2{a}_2{b}_7-{\mu }_2{a}_3{b}_6+{\mu }_2{a}_6{b}_3-{\mu }_2{a}_7{b}_2)e_5\hfill \\ & +& (a_0{b}_6+a_6{b}_0-{\mu }_1{a}_1{b}_7+a_2{b}_4+{\mu }_1{a}_3{b}_5-a_4{b}_2-{\mu }_1{a}_5{b}_3+{\mu }_1{a}_7{b}_1)e_6\hfill \\ & +& (a_0{b}_7+a_7{b}_0-a_1{b}_6+a_2{b}_5+a_3{b}_4-a_4{b}_3-a_5{b}_2+a_6{b}_1)e_7\hfill \end{array}$$

In

Table 2. The multiplication table of Generalized Octonions.

×	1	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_6$	$e_7$
1	1	$e_1$	$e_2$	$e_3$	$e_4$	$e_5$	$e_6$	$e_7$
$e_1$	$e_1$	${\mu }_1$	$e_3$	${\mu }_1{e}_2$	$e_5$	${\mu }_1{e}_4$	$-e_7$	$-{\mu }_1{e}_6$
$e_2$	$e_2$	$-e_3$	${\mu }_2$	$-{\mu }_2{e}_1$	$e_6$	$e_7$	${\mu }_2{e}_4$	${\mu }_2{e}_5$
$e_3$	$e_3$	$-{\mu }_1{e}_2$	${\mu }_2{e}_1$	$-{\mu }_1{\mu }_2$	$e_7$	${\mu }_1{e}_6$	$-{\mu }_2{e}_5$	$-{\mu }_1{\mu }_2{e}_4$
$e_4$	$e_4$	$-e_5$	$-e_6$	$-e_7$	${\mu }_3$	$-{\mu }_3{e}_1$	$-{\mu }_3{e}_2$	$-{\mu }_3{e}_3$
$e_5$	$e_5$	$-{\mu }_1{e}_4$	$-e_7$	$-{\mu }_1{e}_6$	${\mu }_3{e}_1$	$-{\mu }_1{\mu }_3$	${\mu }_3{e}_3$	${\mu }_1{\mu }_3{e}_2$
$e_6$	$e_6$	$e_7$	$-{\mu }_2{e}_4$	${\mu }_2{e}_5$	${\mu }_3{e}_2$	$-{\mu }_3{e}_3$	$-{\mu }_2{\mu }_3$	$-{\mu }_2{\mu }_3{e}_1$
$e_7$	$e_7$	${\mu }_1{e}_6$	$-{\mu }_2{e}_5$	${\mu }_1{\mu }_2{e}_4$	${\mu }_3{e}_3$	$-{\mu }_1{\mu }_3{e}_2$	${\mu }_2{\mu }_3{e}_1$	${\mu }_1{\mu }_2{\mu }_3$

, the multiplication table of Generalized Octonions are shown. The product of $c_1,c_2$ is non-commutative, non-associative, and it is not alternative. But, in special case, the multiplication operation is alternative over real octonions and split octonions. The conjugate of a Generalized Octonion c is:

$$\overline{c}=a_0-\sum _{i=1}^7{a}_i{e}_i$$

which satisfies the followings:

$$\begin{array}{ccc}& & \overline{k_1{c}_1+k_2{c}_2}=k_1\overline{c_1}+k_2\overline{c_2}\hfill \\ & & \overline{c_1\times c_2}=\overline{c_2}\times \overline{c_1}\hfill \\ & & \overline{\overline{c}}=c\hfill \\ & & S_c={\displaystyle \frac{c+\overline{c}}{2}}\hfill \\ & & V_c={\displaystyle \frac{c-\overline{c}}{2}}.\hfill \end{array}$$

The symmetric $\mathbb{R}$-valued bi-linear form of the Generalized Octonion is defined as

$$h(c_1,c_2)={\displaystyle \frac{1}{2}}(c_1\times \overline{c_2}+c_2\times \overline{c_1})$$

or

$$\begin{array}{ccc}\hfill h(c_1,c_2)& =& a_0{b}_0-{\mu }_1{a}_1{b}_1-{\mu }_2{a}_2{b}_2+{\mu }_1{\mu }_2{a}_3{b}_3-{\mu }_3{a}_4{b}_4+{\mu }_1{\mu }_3{a}_5{b}_5+{\mu }_2{\mu }_3{a}_6{b}_6\hfill \\ & & -{\mu }_1{\mu }_2{\mu }_3{a}_7{b}_7.\hfill \end{array}$$

The norm of c is:

$$\left|\right|c\left|\right|=\sqrt{|c\times \overline{c}|}=\sqrt{a_0^2-{\mu }_1{a}_1^2-{\mu }_2{a}_2^2+{\mu }_1{\mu }_2{a}_3^2-{\mu }_3{a}_4^2+{\mu }_1{\mu }_3{a}_5^2+{\mu }_2{\mu }_3{a}_6^2-{\mu }_1{\mu }_2{\mu }_3{a}_7^2}$$

which satisfies the followings:

$$\begin{array}{ccc}& & \left|\right|c\left|\right|=\left|\right|\overline{c}\left|\right|\hfill \\ & & {\left|\right|c\left|\right|}^2=S_c^2+\langle V_c,V_c\rangle \hfill \\ & & \left|\right|c_1\times c_2\left|\right|=\left|\right|c_1\left|\right|.\left|\right|c_2\left|\right|.\hfill \end{array}$$

Note that, the second property satisfies if ${\mu }_1={\mu }_2={\mu }_3=1.$ The inverse of c is:

$$c^{-1}={\displaystyle \frac{\overline{c}}{\left|\right|c\left|\right|}}={\displaystyle \frac{a_0-a_1{e}_1-a_2{e}_2-a_3{e}_3-a_4{e}_4-a_5{e}_5-a_6{e}_6-a_7{e}_7}{\sqrt{a_0^2+{\mu }_1{a}_1^2+{\mu }_2{a}_2^2+{\mu }_1{\mu }_2{a}_3^2+{\mu }_3{a}_4^2+{\mu }_1{\mu }_3{a}_5^2+{\mu }_2{\mu }_3{a}_6^2+{\mu }_1{\mu }_2{\mu }_3{a}_7^2}}},$$

which satisfies the followings:

$$\begin{array}{ccc}& & {\left(kc\right)}^{-1}={\displaystyle \frac{c^{-1}}{k}}\hfill \\ & & {(c_1\times c_2)}^{-1}=c_2^{-1}\times c_1^{-1}.\hfill \end{array}$$

Note that, the second property satisfies if ${\mu }_1={\mu }_2={\mu }_3=1.$ The inner product over $\mathfrak{C}$ is defined as

$$h(c_1,c_2)=a_0{b}_0-{\mu }_1{a}_1{b}_1-{\mu }_2{a}_2{b}_2+{\mu }_1{\mu }_2{a}_3{b}_3-{\mu }_3{a}_4{b}_4+{\mu }_1{\mu }_3{a}_5{b}_5+{\mu }_2{\mu }_3{a}_6{b}_6-{\mu }_1{\mu }_2{\mu }_3{a}_7{b}_7,$$

where $c_1=a_0+{\sum }_{i=1}^7{a}_i{e}_i$ and $c_2=b_0+{\sum }_{i=1}^7{b}_i{e}_i$ which satisfies the following algebraic properties:

$$\begin{array}{ccc}& & h(c_1,c_2)=S_{c_1\times \overline{c_2}}\hfill \\ & & h(c_1,c_2+c_3)=h(c_1,)h(c_1,c_2)+h(c_2,c_3)\hfill \\ & & k.h(c_1,c_2)=h(k{c}_1,c_2)=h(c_1,k{c}_2).\hfill \end{array}$$

Although its definition and a multiplication table for its construction with some of its basic properties such as its norm, are mentioned in the literature, Generalized Octonion have not been researched much since the old years when they were introduced. In fact, Generalized Octonions, which contain many types of octonions, have been studied as real octonions in the literature [5]. The multiplication table has given in an exercise in [4] and the algebraic structure is examined in [21]. In this article, we investigate the classifications of the Generalized Octonions and we give the matrix representations for them. Also we get De Moivre formula for Generalized Octonions. In this way, the $n^{th}$ root theorem is presented. In the final section, we provide an illustrative example for all the results provided in the previous sections. In the Appendix A, we give the Matlab function and codes for all calculations.

2. The Inner Product Space $({\mathbb{R}}^{\mathbf{7}},\langle ,\rangle )$

The inner product space $({\mathbb{R}}^7,\langle ,\rangle )$ is identified with pure Generalized Octonion whose real parts are zero. So, in this section some necessary definitions and some properties about 7-tuples are given.

Definition 1. 

Let $A,B$ are arbitrary 7-tuples in ${\mathbb{R}}^7$ represented as

$$A=(a_1,a_2,a_3,a_4,a_5,a_6,a_7)$$

and

$$B=(b_1,b_2,b_3,b_4,b_5,b_6,b_7).$$

The inner product of them is defined as

$$\langle A,B\rangle ={\mu }_1{a}_1{b}_1+{\mu }_2{a}_2{b}_2-{\mu }_1{\mu }_2{a}_3{b}_3+{\mu }_3{a}_4{b}_4-{\mu }_1{\mu }_3{a}_5{b}_5-{\mu }_2{\mu }_3{a}_6{b}_6+{\mu }_1{\mu }_2{\mu }_3{a}_7{b}_7$$

where ${\mu }_1,{\mu }_2,{\mu }_3\in \mathbb{R}.$

Theorem 1. 

The inner product function $\langle ,\rangle$ satisfies the following properties:

(1) Linearity: $\langle k_{1}A+k_{2}B,C\rangle =k_1\langle A,C\rangle +k_2\langle B,C\rangle .$

(2) Symmetric Property: $\langle A,B\rangle =\langle B,A\rangle$.

(3) Non degenerate Property: When ${\mu }_1,{\mu }_2,{\mu }_3$ are all nonzero real numbers, $\langle A,B\rangle =0$ for all $B\in {\mathbb{R}}^7$, then $A=0$.

Proof. 

Proofs can be easily done by basic substitution.    □

It is important to note that property (3) does not qualify as an inner product in the usual sense, because it lacks positive-definiteness. In particular, the quadratic form $\langle A,A\rangle$ may not be positive for nonzero A. The condition of positive-definiteness has been replaced with the weaker requirement of non-degeneracy, acknowledging that while all positive-definite forms are non-degenerate, not all non-degenerate forms are positive-definite.

Corollary 1. 

With these properties ${\mathbb{R}}^7$ is an inner product space with the inner product defined in Definition 1 if ${\mu }_1,{\mu }_2,{\mu }_3$ are all positive real numbers.

Definition 2. 

Let $A,B$ are arbitrary 7-tuples in ${\mathbb{R}}^7$ represented as

$$A=(a_1,a_2,a_3,a_4,a_5,a_6,a_7)$$

and

$$B=(b_1,b_2,b_3,b_4,b_5,b_6,b_7).$$

The vector product of these two vectors is given as:

$$\begin{array}{ccc}\hfill A▴B& =& \left[\begin{array}{ccccccc}0& {\mu }_2{a}_3& -{\mu }_2{a}_2& {\mu }_3{a}_5& -{\mu }_3{a}_4& {\mu }_2{\mu }_3{a}_7& -{\mu }_2{\mu }_3{a}_6\\ -{\mu }_1{a}_3& 0& {\mu }_1{a}_1& {\mu }_3{a}_6& -{\mu }_1{\mu }_3{a}_7& -{\mu }_3{a}_4& {\mu }_1{\mu }_3{a}_5\\ -a_2& a_1& 0& {\mu }_3{a}_7& -{\mu }_3{a}_6& {\mu }_3{a}_5& -{\mu }_3{a}_4\\ -{\mu }_1{a}_5& -{\mu }_2{a}_6& {\mu }_1{\mu }_2{a}_7& 0& {\mu }_1{a}_1& {\mu }_2{a}_2& -{\mu }_1{\mu }_2{a}_3\\ -a_4& -{\mu }_2{a}_7& {\mu }_2{a}_6& a_1& 0& -{\mu }_2{a}_3& {\mu }_2{a}_2\\ {\mu }_1{a}_7& -a_4& -{\mu }_1{a}_5& a_2& {\mu }_1{a}_3& 0& -{\mu }_1{a}_1\\ a_6& -a_5& -a_4& a_3& a_2& -a_1& 0\end{array}\right]\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\\ b_4\\ b_5\\ b_6\\ b_7\end{array}\right]\hfill \\ & =& ({\mu }_2{a}_3{b}_2-{\mu }_2{a}_2{b}_3+{\mu }_3{a}_5{b}_4-{\mu }_3{a}_4{b}_5+{\mu }_2{\mu }_3{a}_7{b}_6-{\mu }_2{\mu }_3{a}_6{b}_7)e_1\hfill \\ & +& (-{\mu }_1{a}_3{b}_1+{\mu }_1{a}_1{b}_3+{\mu }_3{a}_6{b}_4-{\mu }_1{\mu }_3{a}_7{b}_5-{\mu }_3{a}_4{b}_6+{\mu }_1{\mu }_3{a}_5{b}_7)e_2\hfill \\ & +& (-a_2{b}_1+a_1{b}_2+{\mu }_3{a}_7{b}_4-{\mu }_3{a}_6{b}_5+{\mu }_3{a}_5{b}_6-{\mu }_3{a}_4{b}_7)e_3\hfill \\ & +& (-{\mu }_1{a}_5{b}_1-{\mu }_2{a}_6{b}_2+{\mu }_1{\mu }_2{a}_7{b}_3+{\mu }_1{a}_1{b}_5+{\mu }_2{a}_2{b}_6-{\mu }_1{\mu }_2{a}_3{b}_7)e_4\hfill \\ & +& (-a_4{b}_1-{\mu }_2{a}_7{b}_2+{\mu }_2{a}_6{b}_3+a_1{b}_4-{\mu }_2{a}_3{b}_6+{\mu }_2{a}_2{b}_7)e_5\hfill \\ & +& ({\mu }_1{a}_7{b}_1-a_4{b}_2-{\mu }_1{a}_5{b}_3+a_2{b}_4+{\mu }_1{a}_3{b}_5-{\mu }_1{a}_1{b}_7)e_6\hfill \\ & +& (a_6{b}_1-a_5{b}_2-a_4{b}_3+a_3{b}_4+a_2{b}_5-a_1{b}_6)e_7.\hfill \end{array}$$

Equivalently, the vector product can be stated as:

$$\begin{array}{ccc}\hfill A▴B& =& [{\mu }_2(a_3{b}_2-a_2{b}_3)+{\mu }_3(a_5{b}_4-a_4{b}_5)+{\mu }_2{\mu }_3(a_7{b}_6-a_6{b}_7)]e_1\hfill \\ & +& [{\mu }_1(a_1{b}_3-a_3{b}_1)+{\mu }_3(a_6{b}_4-a_4{b}_6)+{\mu }_1{\mu }_3(a_5{b}_7-a_7{b}_5)]e_2\hfill \\ & +& [(a_1{b}_2-a_2{b}_1)+{\mu }_3(a_5{b}_6-a_6{b}_5+a_7{b}_4-a_4{b}_7)]e_3\hfill \\ & +& [{\mu }_1(a_1{b}_5-a_5{b}_1)+{\mu }_2(a_2{b}_6-a_6{b}_2)+{\mu }_1{\mu }_2(a_7{b}_3-a_3{b}_7)]e_4\hfill \\ & +& [(a_1{b}_4-a_4{b}_1)+{\mu }_2(a_2{b}_7-a_7{b}_2+a_6{b}_3-a_3{b}_6)]e_5\hfill \\ & +& [(a_2{b}_4-a_4{b}_2)+{\mu }_1(a_3{b}_5-a_5{b}_3+a_7{b}_1-a_1{b}_7)]e_6\hfill \\ & +& [(a_2{b}_5-a_5{b}_2)+(a_3{b}_4-a_4{b}_3)+(a_6{b}_1-a_1{b}_6)]e_7\hfill \end{array}$$

Theorem 2. 

The vector product, also known as the cross product, satisfies the following properties:

$$\begin{array}{ccc}& & A▴(B+C)=A▴B+A▴C\hfill \\ & & A▴A=0\hfill \\ & & A▴B=-B▴A\hfill \end{array}$$

Proof. 

Proofs can be easily done by basic substitution.    □

3. Some Properties of Generalized Octonion

In this section, we give some algebraic identities for the Generalized Octonion such as an alternative expression for multiplication, right/left matrix representations, classification with respect to inner product and vector parts.

Theorem 3. 

Let $c_1=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7$ and $c_2=b_0+b_1{e}_1+b_2{e}_2+b_3{e}_3+b_4{e}_4+b_5{e}_5+b_6{e}_6+b_7{e}_7$ be two Generalized Octonion, then

$$c_1\times c_2=S_{c_1}{S}_{c_2}+\langle V_{c_1},V_{c_2}\rangle +S_{c_1}{V}_{c_2}+S_{c_2}{V}_{c_1}+V_{c_1}▴V_{c_2},$$

where $\langle V_{c_1},V_{c_2}\rangle$ is the inner product of vector parts of the $c_1$ and $c_2$ which is given in Definition 1, and $V_{c_1}▴V_{c_2}$ is the vector product of vector parts of the $c_1$ and $c_2$ which is defined in Definition 2.

Proof. 

By substituting the definitions of inner product and vector product in the multiplication of $c_1$ and $c_2$, we can directly prove the theorem.    □

The Generalized Octonion are classified into 3 types when considering the value of the inner product with itself are positive, negative and zero.

$$h(c,c)=a_0^2-V_c^2,$$

where $V_c^2={\mu }_1{a}_1^2+{\mu }_2{a}_2^2-{\mu }_1{\mu }_2{a}_3^2+{\mu }_3{a}_4^2-{\mu }_1{\mu }_3{a}_5^2-{\mu }_2{\mu }_3{a}_6^2+{\mu }_1{\mu }_2{\mu }_3{a}_7^2.$ Also, the Generalized Octonion are classified into 3 types according to the vector part of them. Now, the classifications are given with respect to the sign of the inner product and the vector parts, respectively.

Definition 3. 

Let $c=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7$ be a Generalized Octonion. If $h(c,c)>0$, then c is called a spacelike Generalized Octonion, if $h(c,c)<0$, then c is called a timelike Generalized Octonion and if $h(c,c)=0$, then c is called a lightlike Generalized Octonion.

• When $V_c^2<0$, the Generalized Octonion c is said to have a spacelike vector part.
• When $V_c^2>0$, the Generalized Octonion c is said to have a timelike vector part.
• When $V_c^2=0$, the Generalized Octonion c is said to have a lightlike vector part.

Let $c=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7$ be a Generalized Octonion and $\overrightarrow{u}$ be a unit vector in ${\mathbb{R}}^7$, then there are five classes of the polar representation of c with respect to its inner product and vector part.

(i) If the inner product of c is negative (so, c is a timelike Generalized Octonion), then it is written as

$$c=\left|\right|c\left|\right|.(sinh\alpha +\overrightarrow{u}cosh\alpha ),$$

where

$$sinh\alpha ={\displaystyle \frac{S_c}{\left|\right|c\left|\right|}},cosh\alpha ={\displaystyle \frac{|V_c|}{\left|\right|c\left|\right|}},\overrightarrow{u}={\displaystyle \frac{V_c}{|V_c|}},$$

and $\overrightarrow{u}$ is timelike, Refs [18,19].

(ii) If the inner product of c is positive (so, c is a spacelike Generalized Octonion) and the vector part of c is timelike, then it is written as

$$c=\left|\right|c\left|\right|.(cosh\alpha +\overrightarrow{u}sinh\alpha ),$$

where

$$cosh\alpha ={\displaystyle \frac{S_c}{\left|\right|c\left|\right|}},sinh\alpha ={\displaystyle \frac{|V_c|}{\left|\right|c\left|\right|}},\overrightarrow{u}={\displaystyle \frac{V_c}{|V_c|}},$$

and $\overrightarrow{u}$ is timelike, Refs [18,19].

(iii) If the inner product of c is positive (so, c is a spacelike Generalized Octonion) and the vector part of c is spacelike, then it is written as

$$c=\left|\right|c\left|\right|.(cos\alpha +\overrightarrow{u}sin\alpha ),$$

where

$$cos\alpha ={\displaystyle \frac{S_c}{\left|\right|c\left|\right|}},sin\alpha ={\displaystyle \frac{|V_c|}{\left|\right|c\left|\right|}},\overrightarrow{u}={\displaystyle \frac{V_c}{|V_c|}},$$

and $\overrightarrow{u}$ is spacelike, Refs [18,19].

(iv) A Generalized Octonion c with an inner product of zero (referred to as a lightlike Generalized Octonion) can be expressed in the form as:

$$c=S_c(1+\overrightarrow{u}),$$

where

$$S_c^2=V_c^2,$$

and $\overrightarrow{u}={\displaystyle \frac{V_c}{S_c}}$ is a unit vector in ${\mathbb{R}}^7$ satisfying $(\overrightarrow{u}{)}^2=1.$ Given that:

$$c=S_c+V_c=S_c(1+{\displaystyle \frac{V_c}{S_c}}),$$

and since $S_c^2=V_c^2,$ the condition:

$${\left(\overrightarrow{u}\right)}^2={\left({\displaystyle \frac{V_c}{S_c}}\right)}^2=1$$

is satisfied, confirming that $\overrightarrow{u}$ is indeed a unit vector.

(v) If c is a Generalized Octonion with the lightlike vector part, then

$$c=\left|\right|c\left|\right|.(sgn\left(S_c\right)+\overrightarrow{u}),$$

where

$$\overrightarrow{u}={\displaystyle \frac{V_c}{\left|\right|c\left|\right|}}$$

is a unit vector in ${\mathbb{R}}^7$ with $(\overrightarrow{u}{)}^2=0.$ Note that the vector part of c is lightlike, so $V_c^2=0$, can be determined as

$$\begin{array}{ccc}\hfill c& =& a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7\hfill \\ & =& \sqrt{S_c^2}(sgn(S_c+{\displaystyle \frac{V_c}{\sqrt{S_c^2}}})).\hfill \end{array}$$

Since $V_c^2=0$, then ${(\overrightarrow{u})}^2={({\displaystyle \frac{V_c}{\sqrt{S_c^2}}})}^2=0.$

In

Table 3. The polar forms of the Generalized Octonions.

Type of the Generalized Octonion	Polar Form	Conditions
c is spacelike and the vector part of c is timelike	$c=\left|\right|c\left|\right|.(cosh\alpha +\overrightarrow{u}sinh\alpha )$	$cosh\alpha ={\displaystyle \frac{S_c}{\left|\right|c\left|\right|}},sinh\alpha ={\displaystyle \frac{|V_c|}{\left|\right|c\left|\right|}},\overrightarrow{u}={\displaystyle \frac{V_c}{|V_c|}}$
c is spacelike and the vector part of c is spacelike	$c=\left|\right|c\left|\right|.(cos\alpha +\overrightarrow{u}sin\alpha )$	$cos\alpha ={\displaystyle \frac{S_c}{\left|\right|c\left|\right|}},sin\alpha ={\displaystyle \frac{|V_c|}{\left|\right|c\left|\right|}},\overrightarrow{u}={\displaystyle \frac{V_c}{|V_c|}}$
c is lightlike	$c=S_c(1+\overrightarrow{u})$	$S_c^2=V_c^2$, $\overrightarrow{u}={\displaystyle \frac{V_c}{S_c}}$
The vector part of c is lightlike	$c=\left|\right|c\left|\right|.(sgn\left(S_c\right)+\overrightarrow{u})$	$\overrightarrow{u}={\displaystyle \frac{V_c}{\left|\right|c\left|\right|}}$

, the polar forms of Generalized Octonions are shown.

4. De Moivre’s Formulas of Generalized Octonion

This section introduces De Moivre’s formulas, which provide a method for finding the powers and roots of Generalized Octonions through their polar representations.

Theorem 4. 

Let c be a unit Generalized Octonion in polar form $c=sinh\alpha +\overrightarrow{u}cosh\alpha$ and its inner product be negative. Then, we get

$$c^n=sinhn\alpha +\overrightarrow{u}coshn\alpha$$

if n is an odd integer, and

$$c^n=coshn\alpha +\overrightarrow{u}sinhn\alpha$$

if n is an even integer.

Consider the unit Generalized Octonion $c=cosh\alpha +\overrightarrow{u}sinh\alpha ,$ with a positive inner product, implying that its vector part is timelike. Then, we obtain

$$c^n=coshn\alpha +\overrightarrow{u}sinhn\alpha$$

for all integers n.

Consider the unit Generalized Octonion $c=cos\alpha +\overrightarrow{u}sin\alpha$, with a positive inner product, implying that its vector part is spacelike. Then, we have

$$c^n=cosn\alpha +\overrightarrow{u}sinn\alpha$$

for all $n\in \mathbb{Z},Refs$ [18,19].

Theorem 5. 

Let consider the polar form of a Generalized Octonion $c=S_c(1+\overrightarrow{u})$ whose inner product is zero. Then, we get

$$c^n=S_c^n{2}^{n-1}(1+\overrightarrow{u}),$$

for all integers $n.$

Proof. 

We use the mathematical induction over n. We assume that c is a Generalized Octonion whose inner product is zero and $c^n=S_c^n{2}^{n-1}(1+\overrightarrow{u})$ satisfies. Then, ${\overrightarrow{u}}^2=1$ and

$$\begin{array}{ccc}\hfill c^{n+1}& =& S_c^n{2}^{n-1}(1+\overrightarrow{u})S_c(1+\overrightarrow{u}),\hfill \\ & =& S_c^{n+1}{2}^{n-1}{(1+\overrightarrow{u})}^2,\hfill \\ & =& S_c^{n+1}{2}^n(1+\overrightarrow{u}).\hfill \end{array}$$

   □

Theorem 6. 

Let $c=sgn\left(S_c\right)+\overrightarrow{u}$ be a unit Generalized Octonion whose vector part is lightlike. Then, we obtain

$$c^n=sgn{\left(S_c\right)}^n+n{S}_c^{n-1}\overrightarrow{u},$$

for all integers $n.$

Proof. 

From the mathematical induction over n, we get the proof because the fact ${\overrightarrow{u}}^2=0$.    □

In

Table 4. De Moivre’s formulas for the Generalized Octonions.

The Polar Form	n Is Even	n Is Odd
$c=sinh\alpha +\overrightarrow{u}cosh\alpha$	$c^n=coshn\alpha +\overrightarrow{u}sinhn\alpha$	$c^n=sinhn\alpha +\overrightarrow{u}coshn\alpha$
$c=cosh\alpha +\overrightarrow{u}sinh\alpha$	$c^n=coshn\alpha +\overrightarrow{u}sinhn\alpha$	$c^n=coshn\alpha +\overrightarrow{u}sinhn\alpha$
$c=cos\alpha +\overrightarrow{u}sin\alpha$	$c^n=cosn\alpha +\overrightarrow{u}sinn\alpha$	$c^n=cosn\alpha +\overrightarrow{u}sinn\alpha$
$c=S_c(1+\overrightarrow{u})$	$c^n=S_c^n{2}^{n-1}(1+\overrightarrow{u})$	$c^n=S_c^n{2}^{n-1}(1+\overrightarrow{u})$
$c=sgn\left(S_c\right)+\overrightarrow{u}$	$c^n=sgn{\left(S_c\right)}^n+n{S}_c^{n-1}\overrightarrow{u}$	$c^n=sgn{\left(S_c\right)}^n+n{S}_c^{n-1}\overrightarrow{u}$

, the De Moivre’s formulas for the Generalized Octonions are shown.

5. ${\mathit{n}}^{\mathit{th}}$ Roots of Generalized Octonion

In this section, for any integer n, the $n^{th}$ roots of a Generalized Octonion or in other words the solutions of the equation $x^n-c=0$ is found.

Theorem 7. 

Let $c=sinh\alpha +\overrightarrow{u}cosh\alpha$ be a unit Generalized Octonion in polar form whose inner product is negative.

(i) For even values of n, the equation $x^n=c$ does not have a root.

(ii) For odd values of n, there exist exactly one root of the equation $x^n=c$ calculated as

$$x_0=sinh\gamma +\overrightarrow{u}cosh\gamma ,$$

where $\gamma ={\displaystyle \frac{\alpha }{n}}.$

Let $c=cosh\alpha +\overrightarrow{u}sinh\alpha$ be a Generalized Octonion and unit where the inner product $V_c^2>0$ (positive), indicating that the vector part is timelike.

(i) When n is even, there are four distinct roots of the equation $x^n=c$ and they are given as follows:

$$x_0=sinh\gamma +\overrightarrow{u}cosh\gamma ,$$

(1)

$$x_1=-(sinh\gamma +\overrightarrow{u}cosh\gamma ),$$

(2)

$$x_2=cosh\gamma +\overrightarrow{u}sinh\gamma ,$$

(3)

$$x_3=-(cosh\gamma +\overrightarrow{u}sinh\gamma ),$$

(4)

where $\gamma ={\displaystyle \frac{\alpha }{n}}.$

(ii) When n is odd, there exists exactly one root of the equation $x^n=c$ and the root is

$$x_0=sinh\gamma +\overrightarrow{u}cosh\gamma ,$$

(5)

where $\gamma ={\displaystyle \frac{\alpha }{n}}.$

Let $c=cos\alpha +\overrightarrow{u}sin\alpha$ be a Generalized Octonion an unit where the inner product $V_c^2>0$ (positive), indicating that the vector part is spacelike. Then, there exist n distinct roots as in the form

$$x_p=cos\gamma +\overrightarrow{u}sin\gamma ,$$

(6)

where $\gamma ={\displaystyle \frac{\alpha +2s\pi }{n}},s=0,1,...,n-1.$

Note that in special cases, the Equations (3), (4) and (6) are given as propositions in [18].

Theorem 8. 

Let $c=S_c(1+\overrightarrow{u})$ be a lightlike Generalized Octonion where the inner product of c be zero. Then,

(i) If n is even, then $x^n=c$ has two distinct roots are as follows:

$$\begin{array}{c}\hfill x_0=\eta (1+\overrightarrow{u}),\\ \hfill x_1=-\eta (1+\overrightarrow{u}),\end{array}$$

where $\eta ={\displaystyle \frac{\sqrt[n]{2S_c}}{2}}$.

(ii) If n is odd, then the exactly one root of $x^n=c$ is

$$\begin{array}{c}\hfill x_0=\eta (1+\overrightarrow{u}),\end{array}$$

where $\eta ={\displaystyle \frac{\sqrt[n]{2S_c}}{2}}.$

Proof. 

The proof can be achieved, easily.    □

Theorem 9. 

Let $c=sgn\left(S_c\right)+\overrightarrow{u}$ be a unit Generalized Octonion with the lightlike vector part.

(i) If n is even, then $x^n=c$ has two distinct roots for $sgn\left(S_c\right)>0.$ The roots are as follows:

$$\begin{array}{c}\hfill x_0=\sqrt[n]{S_c}(1+{\displaystyle \frac{\overrightarrow{u}}{n}}),\\ \hfill x_1=-\sqrt[n]{S_c}(1+{\displaystyle \frac{\overrightarrow{u}}{n}}).\end{array}$$

But, the equation $x^n=c$ has no roots for $sgn\left(S_c\right)<0.$

(ii) If n is odd, then $x^n=c$ has exactly one root which is in the following form:

$$\begin{array}{c}\hfill x_0=\sqrt[n]{S_c}(1+{\displaystyle \frac{\overrightarrow{u}}{n}}).\end{array}$$

Proof. 

One can prove the formula directly.    □

Now, by using the Hamilton’s operators, we introduce the right/left matrix representations of the numbers in $\mathfrak{C}.$ Also, we obtain some identities for the representations.

Definition 4. 

Let $c_1=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7$ and $c_2=b_0+b_1{e}_1+b_2{e}_2+b_3{e}_3+b_4{e}_4+b_5{e}_5+b_6{e}_6+b_7{e}_7$ be two Generalized Octonion, then right matrix representation ${\Phi }^{+}:\mathfrak{C}\to \mathfrak{C}$ and the left matrix representation ${\Phi }^{-}:\mathfrak{C}\to \mathfrak{C}$ are defined as

$${\Phi }^{+}\left(c_1\right)=c_1\times c_2,{\Phi }^{-}\left(c_1\right)=c_2\times c_1,c_2\in \mathfrak{C},$$

respectively, and are also called Hamilton’s operator.

Also, the Hamilton’s operators ${\Phi }^{+},{\Phi }^{-}$ could be represented by the matrices as follows:

$$\begin{array}{c}\hfill {\Phi }^{+}\left(c_1\right)=\left(\begin{array}{cccccccc}{a}_0& {\mu }_1{a}_1& {\mu }_2{a}_2& -{\mu }_1{\mu }_2{a}_3& {\mu }_3{a}_4& -{\mu }_1{\mu }_3{a}_5& -{\mu }_2{\mu }_3{a}_6& {\mu }_1{\mu }_2{\mu }_3{a}_7\\ a_1& a_0& {\mu }_2{a}_3& -{\mu }_2{a}_2& {\mu }_3{a}_5& -{\mu }_3{a}_4& {\mu }_2{\mu }_3{a}_7& -{\mu }_2{\mu }_3{a}_6\\ a_2& -{\mu }_1{a}_3& a_0& {\mu }_1{a}_1& {\mu }_3{a}_6& -{\mu }_1{\mu }_3{a}_7& -{\mu }_3{a}_4& {\mu }_1{\mu }_3{a}_5\\ a_3& -a_2& a_1& a_0& {\mu }_3{a}_7& -{\mu }_3{a}_6& {\mu }_3{a}_5& -{\mu }_3{a}_4\\ a_4& -{\mu }_1{a}_5& -{\mu }_2{a}_6& {\mu }_1{\mu }_2{a}_7& a_0& {\mu }_1{a}_1& {\mu }_2{a}_2& -{\mu }_1{\mu }_2{a}_3\\ a_5& -a_4& -{\mu }_2{a}_7& {\mu }_2{a}_6& a_1& a_0& -{\mu }_2{a}_3& {\mu }_2{a}_2\\ a_6& {\mu }_1{a}_7& -a_4& -{\mu }_1{a}_5& a_2& {\mu }_1{a}_3& a_0& -{\mu }_1{a}_1\\ a_7& a_6& -a_5& -a_4& a_3& a_2& -a_1& a_0\end{array}\right).\end{array}$$

In order to obtain the representation matrix, any Generalized Octonion $c_1=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7$ is multiplied from the right side by $1,e_1,e_2,e_3,e_4,e_5,e_6,e_7$, respectively, and the coefficients of the equations in each row correspond to the elements in the columns of the matrix. Similarly, by multiplying $c_1$ from the left side by the Cayley units, we get the following matrix.

$$\begin{array}{c}\hfill {\Phi }^{-}\left(c_1\right)=\left(\begin{array}{cccccccc}{a}_0& {\mu }_1{a}_1& {\mu }_2{a}_2& -{\mu }_1{\mu }_2{a}_3& {\mu }_3{a}_4& -{\mu }_1{\mu }_3{a}_5& -{\mu }_2{\mu }_3{a}_6& {\mu }_1{\mu }_2{\mu }_3{a}_7\\ a_1& a_0& -{\mu }_2{a}_3& {\mu }_2{a}_2& -{\mu }_3{a}_5& {\mu }_3{a}_4& -{\mu }_2{\mu }_3{a}_7& {\mu }_2{\mu }_3{a}_6\\ a_2& {\mu }_1{a}_3& a_0& -{\mu }_1{a}_1& -{\mu }_3{a}_6& {\mu }_1{\mu }_3{a}_7& {\mu }_3{a}_4& -{\mu }_1{\mu }_3{a}_5\\ a_3& a_2& a_1& a_0& -{\mu }_3{a}_7& {\mu }_3{a}_6& -{\mu }_3{a}_5& {\mu }_3{a}_4\\ a_4& {\mu }_1{a}_5& {\mu }_2{a}_6& -{\mu }_1{\mu }_2{a}_7& a_0& -{\mu }_1{a}_1& -{\mu }_2{a}_2& {\mu }_1{\mu }_2{a}_3\\ a_5& a_4& {\mu }_2{a}_7& -{\mu }_2{a}_6& -a_1& a_0& {\mu }_2{a}_3& -{\mu }_2{a}_2\\ a_6& {\mu }_1{a}_7& a_4& {\mu }_1{a}_5& -a_2& -{\mu }_1{a}_3& a_0& {\mu }_1{a}_1\\ a_7& -a_6& a_5& a_4& -a_3& -a_2& a_1& a_0\end{array}\right).\end{array}$$

Theorem 10. 

The functions ${\Phi }^{+}$ and ${\Phi }^{-}$ are bijective.

Proof. 

Let $c=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_4+a_5{e}_5+a_6{e}_6+a_7{e}_7$ be a Generalized Octonion and the function ${\Phi }^{+}:\mathfrak{C}\to \mathfrak{C}$ be defined as above. Then, the kernel of ${\Phi }^{+}$ is

$$ker{\Phi }^{+}=\{c:{\Phi }^{+}\left(c\right)=0\}=\left\{0\right\}.$$

So, ${\Phi }^{+}$ is one to one. The image of ${\Phi }^{+}$ is

$$\Im \left({\Phi }^{+}\right)=\left\{{\Phi }^{+}\left(c\right)\right|c\in \mathfrak{C}\}.$$

By restricting the image set, ${\Phi }^{+}:\mathfrak{C}\to {\Phi }^{+}\left(\mathfrak{C}\right)$ is subset of the real matrices with size 8 over $\mathbb{R}.$ So, ${\Phi }^{+}$ is onto. Similarly, it is directly seen that ${\Phi }^{-}$ is bijective, too.    □

Lemma 1. 

Let $c_1$ and $c_2$ be two Generalized Octonions and $k\in \mathbb{R}$. Then,

$$\begin{array}{ccc}& & {\Phi }^{+}\left(c_1\right)={\Phi }^{+}\left(c_2\right)\iff c_2=c_1\iff {\Phi }^{-}\left(c_1\right)={\Phi }^{-}\left(c_2\right)\hfill \\ & & {\Phi }^{+}(c_1+c_2)={\Phi }^{+}\left(c_1\right)+{\Phi }^{+}\left(c_2\right),{\Phi }^{-}(c_1+c_2)={\Phi }^{-}\left(c_1\right)+{\Phi }^{-}\left(c_2\right)\hfill \\ & & {\Phi }^{+}(c_1\times c_2)={\Phi }^{+}\left(c_1\right){\Phi }^{+}\left(c_2\right),{\Phi }^{-}(c_1\times c_2)={\Phi }^{-}\left(c_1\right){\Phi }^{-}\left(c_2\right)\hfill \\ & & {\Phi }^{+}\left(c_1\right){\Phi }^{-}\left(c_2\right)={\Phi }^{-}\left(c_2\right){\Phi }^{+}\left(c_1\right)\hfill \\ & & {\Phi }^{+}\left(k{c}_1\right)={\Phi }^{+}\left(c_{1}k\right)=k{\Phi }^{+}\left(c_1\right),{\Phi }^{-}\left(k{c}_1\right)={\Phi }^{-}\left(c_{1}k\right)=k{\Phi }^{-}\left(c_1\right)\hfill \\ & & {\Phi }^{+}\left(1\right)={\Phi }^{-}\left(1\right)=I_8,theidentitymatrixwithsizeeight\hfill \\ & & {\Phi }^{+}\left(c_1\right)+{\Phi }^{+}\left(\overline{c_1}\right)=2a_0{I}_8,{\Phi }^{-}\left(c_1\right)+{\Phi }^{-}\left(\overline{c_1}\right)=2a_0{I}_8\hfill \\ & & {\Phi }^{+}\left(o_1^{-1}\right)={\left({\Phi }^{+}\right)}^{-1}\left(c_1\right),{\Phi }^{-}\left(o_1^{-1}\right)={\left({\Phi }^{-}\right)}^{-1}\left(c_1\right),where\left|\right|c_1\left|\right|\ne 0\hfill \end{array}$$

Theorem 11. 

Let $c_1,c_2$ be two Generalized Octonions and $\overrightarrow{c_1}=(a_0,a_1,a_2,a_3,a_4,a_5,a_6,a_7)$ denote the real vector representation of $c_1$. Then

$$\begin{array}{ccc}& & \overrightarrow{c_1\times c_2}={\Phi }^{+}\left(c_1\right)\overrightarrow{c_2}\hfill \\ & & \overrightarrow{c_2\times c_1}={\Phi }^{-}\left(c_2\right)\overrightarrow{c_2}\hfill \\ & & \overrightarrow{c_1\times p\times c_2}={\Phi }^{+}\left(c_1\right){\Phi }^{-}\left(c_2\right)\overrightarrow{p}={\Phi }^{-}\left(c_1\right){\Phi }^{+}\left(c_2\right)\overrightarrow{p}\hfill \\ & & {\Phi }^{+}\left(c_1\right){\Phi }^{-}\left(c_2\right)={\Phi }^{-}\left(c_1\right){\Phi }^{+}\left(c_2\right).\hfill \end{array}$$

Proof. 

The real vector representation of $c\in \mathfrak{C}$ can be rewritten in the form

$$\overrightarrow{c}={\Phi }^{+}\left(c\right){\overrightarrow{s}}^T={\Phi }^{-}\left(c\right){\overrightarrow{s}}^T,$$

where $\overrightarrow{s}=(1,0,0,0,0,0,0,0)$ and ${\overrightarrow{s}}^T$ is the transpose of the 8-length vector $\overrightarrow{s}.$ So, we get

$$\begin{array}{ccc}& & \overrightarrow{c_1\times c_2}={\Phi }^{+}(c_1\times c_2){\mathbf{s}}^T={\Phi }^{+}\left(c_1\right){\Phi }^{+}\left(c_2\right){\mathbf{s}}^T={\Phi }^{+}\left(c_1\right)\overrightarrow{c_2}\hfill \\ & & \overrightarrow{c_2\times c_1}={\Phi }^{-}(c_2\times c_1){\mathbf{s}}^T={\Phi }^{-}\left(c_2\right){\Phi }^{-}\left(c_1\right){\mathbf{s}}^T={\Phi }^{-}\left(c_2\right)\overrightarrow{c_1}\hfill \\ & & \overrightarrow{c_1\times p\times c_2}=\overrightarrow{c_1\times (p\times c_2)}={\Phi }^{+}\left(c_1\right)\overrightarrow{(p\times c_2)}={\Phi }^{-}\left(c_1\right){\Phi }^{+}\left(c_2\right)\overrightarrow{p}\hfill \\ & & \overrightarrow{c_2\times p\times c_1}=\overrightarrow{c_2\times (p\times c_1)}={\Phi }^{-}\left(c_2\right)\overrightarrow{(p\times c_1)}={\Phi }^{-}\left(c_2\right){\Phi }^{+}\left(c_1\right)\overrightarrow{p}\hfill \\ & & {\Phi }^{+}\left(c_1\right){\Phi }^{-}\left(c_2\right)={\Phi }^{-}\left(c_1\right){\Phi }^{+}\left(c_2\right).\hfill \end{array}$$

   □

At this part of the article, the powers and roots are given for only the right representation matrix ${\Phi }^{+}\left(c\right).$ Surely, the similar results can be stated for the left representation matrix ${\Phi }^{-}\left(c\right).$

Theorem 12. 

$\left(i\right)$ If $c=sinh\alpha +\overrightarrow{u}cosh\alpha$ is a unit generalized octonion with the inner product having negative value and a unit timelike vector $\overrightarrow{u}$, the right representation of the generalized octonion ${\Phi }^{+}\left(c\right)$ can be written as follows:

$$\begin{array}{c}\hfill {\Phi }^{+}\left(c\right)=\left(\begin{array}{cccccccc}sinh\alpha & {\mu }_1{u}_{1}cosh\alpha & {\mu }_2{u}_{2}cosh\alpha & -{\mu }_1{\mu }_2{u}_{3}cosh\alpha & {\mu }_3{u}_{4}cosh\alpha & -{\mu }_1{\mu }_3{u}_{5}cosh\alpha & -{\mu }_2{\mu }_3{u}_{6}cosh\alpha & {\mu }_1{\mu }_2{\mu }_3{u}_{7}cosh\alpha \\ u_{1}cosh\alpha & sinh\alpha & {\mu }_2{u}_{3}cosh\alpha & -{\mu }_2{u}_{2}cosh\alpha & {\mu }_3{u}_{5}cosh\alpha & -{\mu }_3{u}_{4}cosh\alpha & {\mu }_2{\mu }_3{u}_{7}cosh\alpha & -{\mu }_2{\mu }_3{u}_{6}cosh\alpha \\ u_{2}cosh\alpha & -{\mu }_1{u}_{3}cosh\alpha & sinh\alpha & {\mu }_1{u}_{1}cosh\alpha & {\mu }_3{u}_{6}cosh\alpha & -{\mu }_1{\mu }_3{u}_{7}cosh\alpha & -{\mu }_3{u}_{4}cosh\alpha & {\mu }_1{\mu }_3{u}_{5}cosh\alpha \\ u_{3}cosh\alpha & -u_{2}cosh\alpha & u_{1}cosh\alpha & sinh\alpha & {\mu }_3{u}_{7}cosh\alpha & -{\mu }_3{u}_{6}cosh\alpha & {\mu }_3{u}_{5}cosh\alpha & -{\mu }_3{u}_{4}cosh\alpha \\ u_{4}cosh\alpha & -{\mu }_1{u}_{5}cosh\alpha & -{\mu }_2{u}_{6}cosh\alpha & {\mu }_1{\mu }_2{u}_{7}cosh\alpha & sinh\alpha & {\mu }_1{u}_{1}cosh\alpha & {\mu }_2{u}_{2}cosh\alpha & -{\mu }_1{\mu }_2{u}_{3}cosh\alpha \\ u_{5}cosh\alpha & -u_{4}cosh\alpha & -{\mu }_2{u}_{7}cosh\alpha & {\mu }_2{u}_{6}cosh\alpha & u_{1}cosh\alpha & sinh\alpha & -{\mu }_2{u}_{3}cosh\alpha & {\mu }_2{u}_{2}cosh\alpha \\ u_{6}cosh\alpha & {\mu }_1{u}_{7}cosh\alpha & -u_{4}cosh\alpha & -{\mu }_1{u}_{5}cosh\alpha & u_{2}cosh\alpha & {\mu }_1{u}_{3}cosh\alpha & sinh\alpha & -{\mu }_1{u}_{1}cosh\alpha \\ u_{7}cosh\alpha & u_{6}cosh\alpha & -u_{5}cosh\alpha & -u_{4}cosh\alpha & u_{3}cosh\alpha & u_{2}cosh\alpha & -u_{1}cosh\alpha & sinh\alpha \end{array}\right).\end{array}$$

For any positive integer n, the $n^{th}$ power of the matrix is given as:

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}sinh\beta & {\mu }_1{u}_{1}cosh\beta & {\mu }_2{u}_{2}cosh\beta & -{\mu }_1{\mu }_2{u}_{3}cosh\beta & {\mu }_3{u}_{4}cosh\beta & -{\mu }_1{\mu }_3{u}_{5}cosh\beta & -{\mu }_2{\mu }_3{u}_{6}cosh\beta & {\mu }_1{\mu }_2{\mu }_3{u}_{7}cosh\beta \\ u_{1}cosh\beta & sinh\beta & {\mu }_2{u}_{3}cosh\beta & -{\mu }_2{u}_{2}cosh\beta & {\mu }_3{u}_{5}cosh\beta & -{\mu }_3{u}_{4}cosh\beta & {\mu }_2{\mu }_3{u}_{7}cosh\beta & -{\mu }_2{\mu }_3{u}_{6}cosh\beta \\ u_{2}cosh\beta & -{\mu }_1{u}_{3}cosh\beta & sinh\beta & {\mu }_1{u}_{1}cosh\beta & {\mu }_3{u}_{6}cosh\beta & -{\mu }_1{\mu }_3{u}_{7}cosh\beta & -{\mu }_3{u}_{4}cosh\beta & {\mu }_1{\mu }_3{u}_{5}cosh\beta \\ u_{3}cosh\beta & -u_{2}cosh\beta & u_{1}cosh\beta & sinh\beta & {\mu }_3{u}_{7}cosh\beta & -{\mu }_3{u}_{6}cosh\beta & {\mu }_3{u}_{5}cosh\beta & -{\mu }_3{u}_{4}cosh\beta \\ u_{4}cosh\beta & -{\mu }_1{u}_{5}cosh\beta & -{\mu }_2{u}_{6}cosh\beta & {\mu }_1{\mu }_2{u}_{7}cosh\beta & sinh\beta & {\mu }_1{u}_{1}cosh\beta & {\mu }_2{u}_{2}cosh\beta & -{\mu }_1{\mu }_2{u}_{3}cosh\beta \\ u_{5}cosh\beta & -u_{4}cosh\beta & -{\mu }_2{u}_{7}cosh\beta & {\mu }_2{u}_{6}cosh\beta & u_{1}cosh\beta & sinh\beta & -{\mu }_2{u}_{3}cosh\beta & {\mu }_2{u}_{2}cosh\beta \\ u_{6}cosh\beta & {\mu }_1{u}_{7}cosh\beta & -u_{4}cosh\beta & -{\mu }_1{u}_{5}cosh\beta & u_{2}cosh\beta & {\mu }_1{u}_{3}cosh\beta & sinh\beta & -{\mu }_1{u}_{1}cosh\beta \\ u_{7}cosh\beta & u_{6}cosh\beta & -u_{5}cosh\beta & -u_{4}cosh\beta & u_{3}cosh\beta & u_{2}cosh\beta & -u_{1}cosh\beta & sinh\beta \end{array}\right),\end{array}$$

where $\beta =n\alpha .$

$\left(ii\right)$ If $c=cosh\alpha +\overrightarrow{u}sinh\alpha$ is a unit generalized octonion with the inner product having positive value and a unit timelike vector $\overrightarrow{u}$, the right representation of the generalized octonion ${\Phi }^{+}\left(c\right)$ can be written as follows:

$$\begin{array}{c}\hfill {\Phi }^{+}\left(c\right)=\left(\begin{array}{cccccccc}cosh\alpha & {\mu }_1{u}_{1}sinh\alpha & {\mu }_2{u}_{2}sinh\alpha & -{\mu }_1{\mu }_2{u}_{3}sinh\alpha & {\mu }_3{u}_{4}sinh\alpha & -{\mu }_1{\mu }_3{u}_{5}sinh\alpha & -{\mu }_2{\mu }_3{u}_{6}sinh\alpha & {\mu }_1{\mu }_2{\mu }_3{u}_{7}sinh\alpha \\ u_{1}sinh\alpha & cosh\alpha & {\mu }_2{u}_{3}sinh\alpha & -{\mu }_2{u}_{2}sinh\alpha & {\mu }_3{u}_{5}sinh\alpha & -{\mu }_3{u}_{4}sinh\alpha & {\mu }_2{\mu }_3{u}_{7}sinh\alpha & -{\mu }_2{\mu }_3{u}_{6}sinh\alpha \\ u_{2}sinh\alpha & -{\mu }_1{u}_{3}sinh\alpha & cosh\alpha & {\mu }_1{u}_{1}sinh\alpha & {\mu }_3{u}_{6}sinh\alpha & -{\mu }_1{\mu }_3{u}_{7}sinh\alpha & -{\mu }_3{u}_{4}sinh\alpha & {\mu }_1{\mu }_3{u}_{5}sinh\alpha \\ u_{3}sinh\alpha & -u_{2}sinh\alpha & u_{1}sinh\alpha & cosh\alpha & {\mu }_3{u}_{7}sinh\alpha & -{\mu }_3{u}_{6}sinh\alpha & {\mu }_3{u}_{5}sinh\alpha & -{\mu }_3{u}_{4}sinh\alpha \\ u_{4}sinh\alpha & -{\mu }_1{u}_{5}sinh\alpha & -{\mu }_2{u}_{6}sinh\alpha & {\mu }_1{\mu }_2{u}_{7}sinh\alpha & cosh\alpha & {\mu }_1{u}_{1}sinh\alpha & {\mu }_2{u}_{2}sinh\alpha & -{\mu }_1{\mu }_2{u}_{3}sinh\alpha \\ u_{5}sinh\alpha & -u_{4}sinh\alpha & -{\mu }_2{u}_{7}sinh\alpha & {\mu }_2{u}_{6}sinh\alpha & u_{1}sinh\alpha & cosh\alpha & -{\mu }_2{u}_{3}sinh\alpha & {\mu }_2{u}_{2}sinh\alpha \\ u_{6}sinh\alpha & {\mu }_1{u}_{7}sinh\alpha & -u_{4}sinh\alpha & -{\mu }_1{u}_{5}sinh\alpha & u_{2}sinh\alpha & {\mu }_1{u}_{3}sinh\alpha & cosh\alpha & -{\mu }_1{u}_{1}sinh\alpha \\ u_{7}sinh\alpha & u_{6}sinh\alpha & -u_{5}sinh\alpha & -u_{4}sinh\alpha & u_{3}sinh\alpha & u_{2}sinh\alpha & -u_{1}sinh\alpha & cosh\alpha \end{array}\right).\end{array}$$

For any positive integer n, the $n^{th}$ power of the matrix is given as:

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}cosh\beta & {\mu }_1{u}_{1}sinh\beta & {\mu }_2{u}_{2}sinh\beta & -{\mu }_1{\mu }_2{u}_{3}sinh\beta & {\mu }_3{u}_{4}sinh\beta & -{\mu }_1{\mu }_3{u}_{5}sinh\beta & -{\mu }_2{\mu }_3{u}_{6}sinh\beta & {\mu }_1{\mu }_2{\mu }_3{u}_{7}sinh\beta \\ u_{1}sinh\beta & cosh\beta & {\mu }_2{u}_{3}sinh\beta & -{\mu }_2{u}_{2}sinh\beta & {\mu }_3{u}_{5}sinh\beta & -{\mu }_3{u}_{4}sinh\beta & {\mu }_2{\mu }_3{u}_{7}sinh\beta & -{\mu }_2{\mu }_3{u}_{6}sinh\beta \\ u_{2}sinh\beta & -{\mu }_1{u}_{3}sinh\beta & cosh\beta & {\mu }_1{u}_{1}sinh\beta & {\mu }_3{u}_{6}sinh\beta & -{\mu }_1{\mu }_3{u}_{7}sinh\beta & -{\mu }_3{u}_{4}sinh\beta & {\mu }_1{\mu }_3{u}_{5}sinh\beta \\ u_{3}sinh\beta & -u_{2}sinh\beta & u_{1}sinh\beta & cosh\beta & {\mu }_3{u}_{7}sinh\beta & -{\mu }_3{u}_{6}sinh\beta & {\mu }_3{u}_{5}sinh\beta & -{\mu }_3{u}_{4}sinh\beta \\ u_{4}sinh\beta & -{\mu }_1{u}_{5}sinh\beta & -{\mu }_2{u}_{6}sinh\beta & {\mu }_1{\mu }_2{u}_{7}sinh\beta & cosh\beta & {\mu }_1{u}_{1}sinh\beta & {\mu }_2{u}_{2}sinh\beta & -{\mu }_1{\mu }_2{u}_{3}sinh\beta \\ u_{5}sinh\beta & -u_{4}sinh\beta & -{\mu }_2{u}_{7}sinh\beta & {\mu }_2{u}_{6}sinh\beta & u_{1}sinh\beta & cosh\beta & -{\mu }_2{u}_{3}sinh\beta & {\mu }_2{u}_{2}sinh\beta \\ u_{6}sinh\beta & {\mu }_1{u}_{7}sinh\beta & -u_{4}sinh\beta & -{\mu }_1{u}_{5}sinh\beta & u_{2}sinh\beta & {\mu }_1{u}_{3}sinh\beta & cosh\beta & -{\mu }_1{u}_{1}sinh\beta \\ u_{7}sinh\beta & u_{6}sinh\beta & -u_{5}sinh\beta & -u_{4}sinh\beta & u_{3}sinh\beta & u_{2}sinh\beta & -u_{1}sinh\beta & cosh\beta \end{array}\right),\end{array}$$

where $\beta =n\alpha .$

$\left(iii\right)$ If $c=cos\alpha +\overrightarrow{u}sin\alpha$ is a unit generalized octonion with the inner product having positive value and a unit spacelike vector $\overrightarrow{u}$, the right representation of the generalized octonion ${\Phi }^{+}\left(c\right)$ can be written as follows:

$$\begin{array}{c}\hfill {\Phi }^{+}\left(c\right)=\left(\begin{array}{cccccccc}cos\alpha & {\mu }_1{u}_{1}sin\alpha & {\mu }_2{u}_{2}sin\alpha & -{\mu }_1{\mu }_2{u}_{3}sin\alpha & {\mu }_3{u}_{4}sin\alpha & -{\mu }_1{\mu }_3{u}_{5}sin\alpha & -{\mu }_2{\mu }_3{u}_{6}sin\alpha & {\mu }_1{\mu }_2{\mu }_3{u}_{7}sin\alpha \\ u_{1}sin\alpha & cos\alpha & {\mu }_2{u}_{3}sin\alpha & -{\mu }_2{u}_{2}sin\alpha & {\mu }_3{u}_{5}sin\alpha & -{\mu }_3{u}_{4}sin\alpha & {\mu }_2{\mu }_3{u}_{7}sin\alpha & -{\mu }_2{\mu }_3{u}_{6}sin\alpha \\ u_{2}sin\alpha & -{\mu }_1{u}_{3}sin\alpha & cos\alpha & {\mu }_1{u}_{1}sin\alpha & {\mu }_3{u}_{6}sin\alpha & -{\mu }_1{\mu }_3{u}_{7}sin\alpha & -{\mu }_3{u}_{4}sin\alpha & {\mu }_1{\mu }_3{u}_{5}sin\alpha \\ u_{3}sin\alpha & -u_{2}sin\alpha & u_{1}sin\alpha & cos\alpha & {\mu }_3{u}_{7}sin\alpha & -{\mu }_3{u}_{6}sin\alpha & {\mu }_3{u}_{5}sin\alpha & -{\mu }_3{u}_{4}sin\alpha \\ u_{4}sin\alpha & -{\mu }_1{u}_{5}sin\alpha & -{\mu }_2{u}_{6}sin\alpha & {\mu }_1{\mu }_2{u}_{7}sin\alpha & cos\alpha & {\mu }_1{u}_{1}sin\alpha & {\mu }_2{u}_{2}sin\alpha & -{\mu }_1{\mu }_2{u}_{3}sin\alpha \\ u_{5}sin\alpha & -u_{4}sin\alpha & -{\mu }_2{u}_{7}sin\alpha & {\mu }_2{u}_{6}sin\alpha & u_{1}sin\alpha & cos\alpha & -{\mu }_2{u}_{3}sin\alpha & {\mu }_2{u}_{2}sin\alpha \\ u_{6}sin\alpha & {\mu }_1{u}_{7}sin\alpha & -u_{4}sin\alpha & -{\mu }_1{u}_{5}sin\alpha & u_{2}sin\alpha & {\mu }_1{u}_{3}sin\alpha & cos\alpha & -{\mu }_1{u}_{1}sin\alpha \\ u_{7}sin\alpha & u_{6}sin\alpha & -u_{5}sin\alpha & -u_{4}sin\alpha & u_{3}sin\alpha & u_{2}sin\alpha & -u_{1}sin\alpha & cos\alpha \end{array}\right).\end{array}$$

For any positive integer n, the $n^{th}$ power of the matrix is given as:

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}cos\left(n\alpha \right)& {\mu }_1{u}_{1}sin\left(n\alpha \right)& {\mu }_2{u}_{2}sin\left(n\alpha \right)& -{\mu }_1{\mu }_2{u}_{3}sin\left(n\alpha \right)& {\mu }_3{u}_{4}sin\left(n\alpha \right)& -{\mu }_1{\mu }_3{u}_{5}sin\left(n\alpha \right)& -{\mu }_2{\mu }_3{u}_{6}sin\left(n\alpha \right)& {\mu }_1{\mu }_2{\mu }_3{u}_{7}sin\left(n\alpha \right)\\ u_{1}sin\left(n\alpha \right)& cos\left(n\alpha \right)& {\mu }_2{u}_{3}sin\left(n\alpha \right)& -{\mu }_2{u}_{2}sin\left(n\alpha \right)& {\mu }_3{u}_{5}sin\left(n\alpha \right)& -{\mu }_3{u}_{4}sin\left(n\alpha \right)& {\mu }_2{\mu }_3{u}_{7}sin\left(n\alpha \right)& -{\mu }_2{\mu }_3{u}_{6}sin\left(n\alpha \right)\\ u_{2}sin\left(n\alpha \right)& -{\mu }_1{u}_{3}sin\left(n\alpha \right)& cos\left(n\alpha \right)& {\mu }_1{u}_{1}sin\left(n\alpha \right)& {\mu }_3{u}_{6}sin\left(n\alpha \right)& -{\mu }_1{\mu }_3{u}_{7}sin\left(n\alpha \right)& -{\mu }_3{u}_{4}sin\left(n\alpha \right)& {\mu }_1{\mu }_3{u}_{5}sin\left(n\alpha \right)\\ u_{3}sin\left(n\alpha \right)& -u_{2}sin\left(n\alpha \right)& u_{1}sin\left(n\alpha \right)& cos\left(n\alpha \right)& {\mu }_3{u}_{7}sin\left(n\alpha \right)& -{\mu }_3{u}_{6}sin\left(n\alpha \right)& {\mu }_3{u}_{5}sin\left(n\alpha \right)& -{\mu }_3{u}_{4}sin\left(n\alpha \right)\\ u_{4}sin\left(n\alpha \right)& -{\mu }_1{u}_{5}sin\left(n\alpha \right)& -{\mu }_2{u}_{6}sin\left(n\alpha \right)& {\mu }_1{\mu }_2{u}_{7}sin\left(n\alpha \right)& cos\left(n\alpha \right)& {\mu }_1{u}_{1}sin\left(n\alpha \right)& {\mu }_2{u}_{2}sin\left(n\alpha \right)& -{\mu }_1{\mu }_2{u}_{3}sin\left(n\alpha \right)\\ u_{5}sin\left(n\alpha \right)& -u_{4}sin\left(n\alpha \right)& -{\mu }_2{u}_{7}sin\left(n\alpha \right)& {\mu }_2{u}_{6}sin\left(n\alpha \right)& u_{1}sin\left(n\alpha \right)& cos\left(n\alpha \right)& -{\mu }_2{u}_{3}sin\left(n\alpha \right)& {\mu }_2{u}_{2}sin\left(n\alpha \right)\\ u_{6}sin\left(n\alpha \right)& {\mu }_1{u}_{7}sin\left(n\alpha \right)& -u_{4}sin\left(n\alpha \right)& -{\mu }_1{u}_{5}sin\left(n\alpha \right)& u_{2}sin\left(n\alpha \right)& {\mu }_1{u}_{3}sin\left(n\alpha \right)& cos\left(n\alpha \right)& -{\mu }_1{u}_{1}sin\left(n\alpha \right)\\ u_{7}sin\left(n\alpha \right)& u_{6}sin\left(n\alpha \right)& -u_{5}sin\left(n\alpha \right)& -u_{4}sin\left(n\alpha \right)& u_{3}sin\left(n\alpha \right)& u_{2}sin\left(n\alpha \right)& -u_{1}sin\left(n\alpha \right)& cos\left(n\alpha \right)\end{array}\right).\end{array}$$

$\left(iv\right)$ If $c=S_c(1+\overrightarrow{u})$ is a unit Generalized Octonion with a zero inner product (i.e., a lightlike Generalized Octonion), the right representation of the Generalized Octonion ${\Phi }^{+}\left(c\right)$ can be formulated as follows:

$$\begin{array}{c}\hfill {\Phi }^{+}\left(c\right)=\left(\begin{array}{cccccccc}{S}_c& {\mu }_1{u}_1& {\mu }_2{u}_2& -{\mu }_1{\mu }_2{u}_3& {\mu }_3{u}_4& -{\mu }_1{\mu }_3{u}_5& -{\mu }_2{\mu }_3{u}_6& {\mu }_1{\mu }_2{\mu }_3{u}_7\\ u_1& S_c& {\mu }_2{u}_3& -{\mu }_2{u}_2& {\mu }_3{u}_5& -{\mu }_3{u}_4& {\mu }_2{\mu }_3{u}_7& -{\mu }_2{\mu }_3{u}_6\\ u_2& -{\mu }_1{u}_3& S_c& {\mu }_1{u}_1& {\mu }_3{u}_6& -{\mu }_1{\mu }_3{u}_7& -{\mu }_3{u}_4& {\mu }_1{\mu }_3{u}_5\\ u_3& -u_2& u_1& S_c& {\mu }_3{u}_7& -{\mu }_3{u}_6& {\mu }_3{u}_5& -{\mu }_3{u}_4\\ u_4& -{\mu }_1{u}_5& -{\mu }_2{u}_6& {\mu }_1{\mu }_2{u}_7& S_c& {\mu }_1{u}_1& {\mu }_2{u}_2& -{\mu }_1{\mu }_2{u}_3\\ u_5& -u_4& -{\mu }_2{u}_7& {\mu }_2{u}_6& u_1& S_c& -{\mu }_2{u}_3& {\mu }_2{u}_2\\ u_6& {\mu }_1{u}_7& -u_4& -{\mu }_1{u}_5& u_2& {\mu }_1{u}_3& S_c& -{\mu }_1{u}_1\\ u_7& u_6& -u_5& -u_4& u_3& u_2& -u_1& S_c\end{array}\right).\end{array}$$

For any positive integer n, the $n^{th}$ power of the matrix is given as:

$$\begin{array}{c}\hfill S_c^n{2}^{n-1}\left(\begin{array}{cccccccc}1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3{u}_7}{S_c}}\\ {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_2{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}\\ {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}\\ {\displaystyle \frac{u_3}{S_c}}& -{\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}\\ {\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{u}_7}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}\\ {\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_7}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& -{\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}\\ {\displaystyle \frac{u_6}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_7}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& -{\displaystyle \frac{{\mu }_1{u}_1}{S_c}}\\ {\displaystyle \frac{u_7}{S_c}}& {\displaystyle \frac{u_6}{S_c}}& -{\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& {\displaystyle \frac{u_3}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{u_1}{S_c}}& 1\end{array}\right).\end{array}$$

$\left(v\right)$ If $c=sgn\left(S_c\right)+\overrightarrow{u}$ is a unit generalized octonion with the lightlike vector part, the right representation of the Generalized Octonion ${\Phi }^{+}\left(c\right)$ can be depicted as:

$$\begin{array}{c}\hfill {\Phi }^{+}\left(c\right)=\left(\begin{array}{cccccccc}sgn\left(S_c\right)& {\mu }_1{u}_1& {\mu }_2{u}_2& -{\mu }_1{\mu }_2{u}_3& {\mu }_3{u}_4& -{\mu }_1{\mu }_3{u}_5& -{\mu }_2{\mu }_3{u}_6& {\mu }_1{\mu }_2{\mu }_3{u}_7\\ u_1& sgn\left(S_c\right)& {\mu }_2{u}_3& -{\mu }_2{u}_2& {\mu }_3{u}_5& -{\mu }_3{u}_4& {\mu }_2{\mu }_3{u}_7& -{\mu }_2{\mu }_3{u}_6\\ u_2& -{\mu }_1{u}_3& sgn\left(S_c\right)& {\mu }_1{u}_1& {\mu }_3{u}_6& -{\mu }_1{\mu }_3{u}_7& -{\mu }_3{u}_4& {\mu }_1{\mu }_3{u}_5\\ u_3& -u_2& u_1& sgn\left(S_c\right)& {\mu }_3{u}_7& -{\mu }_3{u}_6& {\mu }_3{u}_5& -{\mu }_3{u}_4\\ u_4& -{\mu }_1{u}_5& -{\mu }_2{u}_6& {\mu }_1{\mu }_2{u}_7& sgn\left(S_c\right)& {\mu }_1{u}_1& {\mu }_2{u}_2& -{\mu }_1{\mu }_2{u}_3\\ u_5& -u_4& -{\mu }_2{u}_7& {\mu }_2{u}_6& u_1& sgn\left(S_c\right)& -{\mu }_2{u}_3& {\mu }_2{u}_2\\ u_6& {\mu }_1{u}_7& -u_4& -{\mu }_1{u}_5& u_2& {\mu }_1{u}_3& sgn\left(S_c\right)& -{\mu }_1{u}_1\\ u_7& u_6& -u_5& -u_4& u_3& u_2& -u_1& sgn\left(S_c\right)\end{array}\right).\end{array}$$

For any positive integer n, the $n^{th}$ power of the matrix is demonstrated as:

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}sg{n}^n\left(S_c\right)& n{S}_c^{n-1}{\mu }_1{u}_1& n{S}_c^{n-1}{\mu }_2{u}_2& -n{S}_c^{n-1}{\mu }_1{\mu }_2{u}_3& n{S}_c^{n-1}{\mu }_3{u}_4& -n{S}_c^{n-1}{\mu }_1{\mu }_3{u}_5& -n{S}_c^{n-1}{\mu }_2{\mu }_3{u}_6& n{S}_c^{n-1}{\mu }_1{\mu }_2{\mu }_3{u}_7\\ n{S}_c^{n-1}{u}_1& sg{n}^n\left(S_c\right)& n{S}_c^{n-1}{\mu }_2{u}_3& -n{S}_c^{n-1}{\mu }_2{u}_2& {\mu }_3{u}_5& -n{S}_c^{n-1}{\mu }_3{u}_4& n{S}_c^{n-1}{\mu }_2{\mu }_3{u}_7& -n{S}_c^{n-1}{\mu }_2{\mu }_3{u}_6\\ n{S}_c^{n-1}{u}_2& -n{S}_c^{n-1}{\mu }_1{u}_3& sg{n}^n\left(S_c\right)& n{S}_c^{n-1}{\mu }_1{u}_1& n{S}_c^{n-1}{\mu }_3{u}_6& -n{S}_c^{n-1}{\mu }_1{\mu }_3{u}_7& -n{S}_c^{n-1}{\mu }_3{u}_4& n{S}_c^{n-1}{\mu }_1{\mu }_3{u}_5\\ n{S}_c^{n-1}{u}_3& -n{S}_c^{n-1}{u}_2& n{S}_c^{n-1}{u}_1& sg{n}^n\left(S_c\right)& n{S}_c^{n-1}{\mu }_3{u}_7& -n{S}_c^{n-1}{\mu }_3{u}_6& n{S}_c^{n-1}{\mu }_3{u}_5& -n{S}_c^{n-1}{\mu }_3{u}_4\\ n{S}_c^{n-1}{u}_4& -n{S}_c^{n-1}{\mu }_1{u}_5& -n{S}_c^{n-1}{\mu }_2{u}_6& n{S}_c^{n-1}{\mu }_1{\mu }_2{u}_7& sg{n}^n\left(S_c\right)& n{S}_c^{n-1}{\mu }_1{u}_1& n{S}_c^{n-1}{\mu }_2{u}_2& -n{S}_c^{n-1}{\mu }_1{\mu }_2{u}_3\\ n{S}_c^{n-1}{u}_5& -n{S}_c^{n-1}{u}_4& -n{S}_c^{n-1}{\mu }_2{u}_7& n{S}_c^{n-1}{\mu }_2{u}_6& n{S}_c^{n-1}{u}_1& sg{n}^n\left(S_c\right)& -n{S}_c^{n-1}{\mu }_2{u}_3& n{S}_c^{n-1}{\mu }_2{u}_2\\ n{S}_c^{n-1}{u}_6& n{S}_c^{n-1}{\mu }_1{u}_7& -n{S}_c^{n-1}{u}_4& -n{S}_c^{n-1}{\mu }_1{u}_5& n{S}_c^{n-1}{u}_2& n{S}_c^{n-1}{\mu }_1{u}_3& sg{n}^n\left(S_c\right)& -n{S}_c^{n-1}{\mu }_1{u}_1\\ n{S}_c^{n-1}{u}_7& n{S}_c^{n-1}{u}_6& -n{S}_c^{n-1}{u}_5& -n{S}_c^{n-1}{u}_4& n{S}_c^{n-1}{u}_3& n{S}_c^{n-1}{u}_2& -n{S}_c^{n-1}{u}_1& sg{n}^n\left(S_c\right)\end{array}\right).\end{array}$$

Proof. 

By using the mathematical induction over n, and using the trigonometric identities for summation, the proofs can be derived, clearly.    □

Theorem 13. 

Let ${\Phi }^{+}\left(c\right)$ be the right representation matrix of the generalized octonion c which is unit and has negative inner product and $\overrightarrow{u}$ be a unit timelike vector with length 7. Then, the equation $x^n-c=0$ or $X^n-{\Phi }^{+}\left(c\right)=\mathbf{0}$ has the following solutions:

(i) If n is even, then there is no solution.

(ii) If n is odd, then there is only unique solution such that

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma & {\mu }_3{u}_{4}cosh\gamma & -{\mu }_1{\mu }_3{u}_{5}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{\mu }_3{u}_{7}cosh\gamma \\ u_{1}cosh\gamma & sinh\gamma & {\mu }_2{u}_{3}cosh\gamma & -{\mu }_2{u}_{2}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_2{\mu }_3{u}_{7}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma \\ u_{2}cosh\gamma & -{\mu }_1{u}_{3}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_3{u}_{6}cosh\gamma & -{\mu }_1{\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_1{\mu }_3{u}_{5}cosh\gamma \\ u_{3}cosh\gamma & -u_{2}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & {\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{6}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma \\ u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & -{\mu }_2{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{u}_{7}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma \\ u_{5}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_2{u}_{7}cosh\gamma & {\mu }_2{u}_{6}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & -{\mu }_2{u}_{3}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma \\ u_{6}cosh\gamma & {\mu }_1{u}_{7}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & u_{2}cosh\gamma & {\mu }_1{u}_{3}cosh\gamma & sinh\gamma & -{\mu }_1{u}_{1}cosh\gamma \\ u_{7}cosh\gamma & u_{6}cosh\gamma & -u_{5}cosh\gamma & -u_{4}cosh\gamma & u_{3}cosh\gamma & u_{2}cosh\gamma & -u_{1}cosh\gamma & sinh\gamma \end{array}\right),\end{array}$$

where $\gamma ={\displaystyle \frac{\alpha }{n}}.$

Proof. 

By using the Theorem 12, $n^{th}$ power of the root matrix is given ${\Phi }^{+}\left(c\right)$. So, the theorem is thus proven.    □

Theorem 14. 

Let ${\Phi }^{+}\left(c\right)$ be the right representation matrix of the generalized octonion c which is unit and has positive inner product and $\overrightarrow{u}$ be a unit timelike vector with length 7. Then, the equation $x^n-c=0$ or $X^n-{\Phi }^{+}\left(c\right)=\mathbf{0}$ has the following solutions:

(i) If n is even, then there are $4-$distinct solutions such that

$$\begin{array}{c}\hfill X_0=\left(\begin{array}{cccccccc}sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma & {\mu }_3{u}_{4}cosh\gamma & -{\mu }_1{\mu }_3{u}_{5}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{\mu }_3{u}_{7}cosh\gamma \\ u_{1}cosh\gamma & sinh\gamma & {\mu }_2{u}_{3}cosh\gamma & -{\mu }_2{u}_{2}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_2{\mu }_3{u}_{7}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma \\ u_{2}cosh\gamma & -{\mu }_1{u}_{3}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_3{u}_{6}cosh\gamma & -{\mu }_1{\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_1{\mu }_3{u}_{5}cosh\gamma \\ u_{3}cosh\gamma & -u_{2}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & {\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{6}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma \\ u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & -{\mu }_2{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{u}_{7}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma \\ u_{5}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_2{u}_{7}cosh\gamma & {\mu }_2{u}_{6}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & -{\mu }_2{u}_{3}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma \\ u_{6}cosh\gamma & {\mu }_1{u}_{7}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & u_{2}cosh\gamma & {\mu }_1{u}_{3}cosh\gamma & sinh\gamma & -{\mu }_1{u}_{1}cosh\gamma \\ u_{7}cosh\gamma & u_{6}cosh\gamma & -u_{5}cosh\gamma & -u_{4}cosh\gamma & u_{3}cosh\gamma & u_{2}cosh\gamma & -u_{1}cosh\gamma & sinh\gamma \end{array}\right),\end{array}$$

where $\gamma ={\displaystyle \frac{\alpha }{n}},$

$$\begin{array}{c}\hfill X_1=-\left(\begin{array}{cccccccc}sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma & {\mu }_3{u}_{4}cosh\gamma & -{\mu }_1{\mu }_3{u}_{5}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{\mu }_3{u}_{7}cosh\gamma \\ u_{1}cosh\gamma & sinh\gamma & {\mu }_2{u}_{3}cosh\gamma & -{\mu }_2{u}_{2}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_2{\mu }_3{u}_{7}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma \\ u_{2}cosh\gamma & -{\mu }_1{u}_{3}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_3{u}_{6}cosh\gamma & -{\mu }_1{\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_1{\mu }_3{u}_{5}cosh\gamma \\ u_{3}cosh\gamma & -u_{2}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & {\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{6}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma \\ u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & -{\mu }_2{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{u}_{7}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma \\ u_{5}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_2{u}_{7}cosh\gamma & {\mu }_2{u}_{6}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & -{\mu }_2{u}_{3}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma \\ u_{6}cosh\gamma & {\mu }_1{u}_{7}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & u_{2}cosh\gamma & {\mu }_1{u}_{3}cosh\gamma & sinh\gamma & -{\mu }_1{u}_{1}cosh\gamma \\ u_{7}cosh\gamma & u_{6}cosh\gamma & -u_{5}cosh\gamma & -u_{4}cosh\gamma & u_{3}cosh\gamma & u_{2}cosh\gamma & -u_{1}cosh\gamma & sinh\gamma \end{array}\right),\end{array}$$

where $\gamma ={\displaystyle \frac{\alpha }{n}},$

$$\begin{array}{c}\hfill X_2=\left(\begin{array}{cccccccc}cosh\gamma & {\mu }_1{u}_{1}sinh\gamma & {\mu }_2{u}_{2}sinh\gamma & -{\mu }_1{\mu }_2{u}_{3}sinh\gamma & {\mu }_3{u}_{4}sinh\gamma & -{\mu }_1{\mu }_3{u}_{5}sinh\gamma & -{\mu }_2{\mu }_3{u}_{6}sinh\gamma & {\mu }_1{\mu }_2{\mu }_3{u}_{7}sinh\gamma \\ u_{1}sinh\gamma & cosh\gamma & {\mu }_2{u}_{3}sinh\gamma & -{\mu }_2{u}_{2}sinh\gamma & {\mu }_3{u}_{5}sinh\gamma & -{\mu }_3{u}_{4}sinh\gamma & {\mu }_2{\mu }_3{u}_{7}sinh\gamma & -{\mu }_2{\mu }_3{u}_{6}sinh\gamma \\ u_{2}sinh\gamma & -{\mu }_1{u}_{3}sinh\gamma & cosh\gamma & {\mu }_1{u}_{1}sinh\gamma & {\mu }_3{u}_{6}sinh\gamma & -{\mu }_1{\mu }_3{u}_{7}sinh\gamma & -{\mu }_3{u}_{4}sinh\gamma & {\mu }_1{\mu }_3{u}_{5}sinh\gamma \\ u_{3}sinh\gamma & -u_{2}sinh\gamma & u_{1}sinh\gamma & cosh\gamma & {\mu }_3{u}_{7}sinh\gamma & -{\mu }_3{u}_{6}sinh\gamma & {\mu }_3{u}_{5}sinh\gamma & -{\mu }_3{u}_{4}sinh\gamma \\ u_{4}sinh\gamma & -{\mu }_1{u}_{5}sinh\gamma & -{\mu }_2{u}_{6}sinh\gamma & {\mu }_1{\mu }_2{u}_{7}sinh\gamma & cosh\gamma & {\mu }_1{u}_{1}sinh\gamma & {\mu }_2{u}_{2}sinh\gamma & -{\mu }_1{\mu }_2{u}_{3}sinh\gamma \\ u_{5}sinh\gamma & -u_{4}sinh\gamma & -{\mu }_2{u}_{7}sinh\gamma & {\mu }_2{u}_{6}sinh\gamma & u_{1}sinh\gamma & cosh\gamma & -{\mu }_2{u}_{3}sinh\gamma & {\mu }_2{u}_{2}sinh\gamma \\ u_{6}sinh\gamma & {\mu }_1{u}_{7}sinh\gamma & -u_{4}sinh\gamma & -{\mu }_1{u}_{5}sinh\gamma & u_{2}sinh\gamma & {\mu }_1{u}_{3}sinh\gamma & cosh\gamma & -{\mu }_1{u}_{1}sinh\gamma \\ u_{7}sinh\gamma & u_{6}sinh\gamma & -u_{5}sinh\gamma & -u_{4}sinh\gamma & u_{3}sinh\gamma & u_{2}sinh\gamma & -u_{1}sinh\gamma & cosh\gamma \end{array}\right),\end{array}$$

where $\gamma ={\displaystyle \frac{\alpha }{n}},$

$$\begin{array}{c}\hfill X_3=-\left(\begin{array}{cccccccc}cosh\gamma & {\mu }_1{u}_{1}sinh\gamma & {\mu }_2{u}_{2}sinh\gamma & -{\mu }_1{\mu }_2{u}_{3}sinh\gamma & {\mu }_3{u}_{4}sinh\gamma & -{\mu }_1{\mu }_3{u}_{5}sinh\gamma & -{\mu }_2{\mu }_3{u}_{6}sinh\gamma & {\mu }_1{\mu }_2{\mu }_3{u}_{7}sinh\gamma \\ u_{1}sinh\gamma & cosh\gamma & {\mu }_2{u}_{3}sinh\gamma & -{\mu }_2{u}_{2}sinh\gamma & {\mu }_3{u}_{5}sinh\gamma & -{\mu }_3{u}_{4}sinh\gamma & {\mu }_2{\mu }_3{u}_{7}sinh\gamma & -{\mu }_2{\mu }_3{u}_{6}sinh\gamma \\ u_{2}sinh\gamma & -{\mu }_1{u}_{3}sinh\gamma & cosh\gamma & {\mu }_1{u}_{1}sinh\gamma & {\mu }_3{u}_{6}sinh\gamma & -{\mu }_1{\mu }_3{u}_{7}sinh\gamma & -{\mu }_3{u}_{4}sinh\gamma & {\mu }_1{\mu }_3{u}_{5}sinh\gamma \\ u_{3}sinh\gamma & -u_{2}sinh\gamma & u_{1}sinh\gamma & cosh\gamma & {\mu }_3{u}_{7}sinh\gamma & -{\mu }_3{u}_{6}sinh\gamma & {\mu }_3{u}_{5}sinh\gamma & -{\mu }_3{u}_{4}sinh\gamma \\ u_{4}sinh\gamma & -{\mu }_1{u}_{5}sinh\gamma & -{\mu }_2{u}_{6}sinh\gamma & {\mu }_1{\mu }_2{u}_{7}sinh\gamma & cosh\gamma & {\mu }_1{u}_{1}sinh\gamma & {\mu }_2{u}_{2}sinh\gamma & -{\mu }_1{\mu }_2{u}_{3}sinh\gamma \\ u_{5}sinh\gamma & -u_{4}sinh\gamma & -{\mu }_2{u}_{7}sinh\gamma & {\mu }_2{u}_{6}sinh\gamma & u_{1}sinh\gamma & cosh\gamma & -{\mu }_2{u}_{3}sinh\gamma & {\mu }_2{u}_{2}sinh\gamma \\ u_{6}sinh\gamma & {\mu }_1{u}_{7}sinh\gamma & -u_{4}sinh\gamma & -{\mu }_1{u}_{5}sinh\gamma & u_{2}sinh\gamma & {\mu }_1{u}_{3}sinh\gamma & cosh\gamma & -{\mu }_1{u}_{1}sinh\gamma \\ u_{7}sinh\gamma & u_{6}sinh\gamma & -u_{5}sinh\gamma & -u_{4}sinh\gamma & u_{3}sinh\gamma & u_{2}sinh\gamma & -u_{1}sinh\gamma & cosh\gamma \end{array}\right),\end{array}$$

where $\gamma ={\displaystyle \frac{\alpha }{n}}.$

(ii) If n is odd, then there is unique solution as follows

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma & {\mu }_3{u}_{4}cosh\gamma & -{\mu }_1{\mu }_3{u}_{5}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{\mu }_3{u}_{7}cosh\gamma \\ u_{1}cosh\gamma & sinh\gamma & {\mu }_2{u}_{3}cosh\gamma & -{\mu }_2{u}_{2}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_2{\mu }_3{u}_{7}cosh\gamma & -{\mu }_2{\mu }_3{u}_{6}cosh\gamma \\ u_{2}cosh\gamma & -{\mu }_1{u}_{3}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_3{u}_{6}cosh\gamma & -{\mu }_1{\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma & {\mu }_1{\mu }_3{u}_{5}cosh\gamma \\ u_{3}cosh\gamma & -u_{2}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & {\mu }_3{u}_{7}cosh\gamma & -{\mu }_3{u}_{6}cosh\gamma & {\mu }_3{u}_{5}cosh\gamma & -{\mu }_3{u}_{4}cosh\gamma \\ u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & -{\mu }_2{u}_{6}cosh\gamma & {\mu }_1{\mu }_2{u}_{7}cosh\gamma & sinh\gamma & {\mu }_1{u}_{1}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma & -{\mu }_1{\mu }_2{u}_{3}cosh\gamma \\ u_{5}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_2{u}_{7}cosh\gamma & {\mu }_2{u}_{6}cosh\gamma & u_{1}cosh\gamma & sinh\gamma & -{\mu }_2{u}_{3}cosh\gamma & {\mu }_2{u}_{2}cosh\gamma \\ u_{6}cosh\gamma & {\mu }_1{u}_{7}cosh\gamma & -u_{4}cosh\gamma & -{\mu }_1{u}_{5}cosh\gamma & u_{2}cosh\gamma & {\mu }_1{u}_{3}cosh\gamma & sinh\gamma & -{\mu }_1{u}_{1}cosh\gamma \\ u_{7}cosh\gamma & u_{6}cosh\gamma & -u_{5}cosh\gamma & -u_{4}cosh\gamma & u_{3}cosh\gamma & u_{2}cosh\gamma & -u_{1}cosh\gamma & sinh\gamma \end{array}\right),\end{array}$$

where $\gamma ={\displaystyle \frac{\alpha }{n}}.$

Proof. 

By using the Theorem 12, $n^{th}$ power of the $X_0,X_1,X_2,X_3$ and the given matrix in (ii) are equal ${\Phi }^{+}\left(c\right)$. This completes the proof.    □

Theorem 15. 

Let ${\Phi }^{+}\left(c\right)$ be the right representation matrix of the generalized octonion c which is unit and has positive inner product and $\overrightarrow{u}$ be a unit spacelike vector with length 7. Then, the equation $x^n-c=0$ or $X^n-{\Phi }^{+}\left(c\right)=\mathbf{0}$ has $n-$solutions in the following form

$$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}cos\gamma & {\mu }_1{u}_{1}sin\gamma & {\mu }_2{u}_{2}sin\gamma & -{\mu }_1{\mu }_2{u}_{3}sin\gamma & {\mu }_3{u}_{4}sin\gamma & -{\mu }_1{\mu }_3{u}_{5}sin\gamma & -{\mu }_2{\mu }_3{u}_{6}sin\gamma & {\mu }_1{\mu }_2{\mu }_3{u}_{7}sin\gamma \\ u_{1}sin\gamma & cos\gamma & {\mu }_2{u}_{3}sin\gamma & -{\mu }_2{u}_{2}sin\gamma & {\mu }_3{u}_{5}sin\gamma & -{\mu }_3{u}_{4}sin\gamma & {\mu }_2{\mu }_3{u}_{7}sin\gamma & -{\mu }_2{\mu }_3{u}_{6}sin\gamma \\ u_{2}sin\gamma & -{\mu }_1{u}_{3}sin\gamma & cos\gamma & {\mu }_1{u}_{1}sin\gamma & {\mu }_3{u}_{6}sin\gamma & -{\mu }_1{\mu }_3{u}_{7}sin\gamma & -{\mu }_3{u}_{4}sin\gamma & {\mu }_1{\mu }_3{u}_{5}sin\gamma \\ u_{3}sin\gamma & -u_{2}sin\gamma & u_{1}sin\gamma & cos\gamma & {\mu }_3{u}_{7}sin\gamma & -{\mu }_3{u}_{6}sin\gamma & {\mu }_3{u}_{5}sin\gamma & -{\mu }_3{u}_{4}sin\gamma \\ u_{4}sin\gamma & -{\mu }_1{u}_{5}sin\gamma & -{\mu }_2{u}_{6}sin\gamma & {\mu }_1{\mu }_2{u}_{7}sin\gamma & cos\gamma & {\mu }_1{u}_{1}sin\gamma & {\mu }_2{u}_{2}sin\gamma & -{\mu }_1{\mu }_2{u}_{3}sin\gamma \\ u_{5}sin\gamma & -u_{4}sin\gamma & -{\mu }_2{u}_{7}sin\gamma & {\mu }_2{u}_{6}sin\gamma & u_{1}sin\gamma & cos\gamma & -{\mu }_2{u}_{3}sin\gamma & {\mu }_2{u}_{2}sin\gamma \\ u_{6}sin\gamma & {\mu }_1{u}_{7}sin\gamma & -u_{4}sin\gamma & -{\mu }_1{u}_{5}sin\gamma & u_{2}sin\gamma & {\mu }_1{u}_{3}sin\gamma & cos\gamma & -{\mu }_1{u}_{1}sin\gamma \\ u_{7}sin\gamma & u_{6}sin\gamma & -u_{5}sin\gamma & -u_{4}sin\gamma & u_{3}sin\gamma & u_{2}sin\gamma & -u_{1}sin\gamma & cos\gamma \end{array}\right),\end{array}$$

where $\gamma ={\displaystyle \frac{\alpha +2s\pi }{n}},s=0,1,2,...,n-1.$

Proof. 

The proof is similar to Theorem 13.    □

Theorem 16. 

Let ${\Phi }^{+}\left(c\right)$ be the right representation matrix of the generalized octonion c which has zero inner product and $\overrightarrow{u}$ be a unit vector with length 7. Then, the equation $x^n-c=0$ or $X^n-{\Phi }^{+}\left(c\right)=\mathbf{0}$ has the solutions in the following forms

(i) If n is even, then there are two solutions such that

$$\begin{array}{c}\hfill X_0=\mathfrak{S}\left(\begin{array}{cccccccc}1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3{u}_7}{S_c}}\\ {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_2{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}\\ {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}\\ {\displaystyle \frac{u_3}{S_c}}& -{\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}\\ {\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{u}_7}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}\\ {\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_7}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& -{\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}\\ {\displaystyle \frac{u_6}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_7}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& -{\displaystyle \frac{{\mu }_1{u}_1}{S_c}}\\ {\displaystyle \frac{u_7}{S_c}}& {\displaystyle \frac{u_6}{S_c}}& -{\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& {\displaystyle \frac{u_3}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{u_1}{S_c}}& 1\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill X_1=-\mathfrak{S}\left(\begin{array}{cccccccc}1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3{u}_7}{S_c}}\\ {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_2{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}\\ {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}\\ {\displaystyle \frac{u_3}{S_c}}& -{\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}\\ {\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{u}_7}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}\\ {\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_7}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& -{\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}\\ {\displaystyle \frac{u_6}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_7}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& -{\displaystyle \frac{{\mu }_1{u}_1}{S_c}}\\ {\displaystyle \frac{u_7}{S_c}}& {\displaystyle \frac{u_6}{S_c}}& -{\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& {\displaystyle \frac{u_3}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{u_1}{S_c}}& 1\end{array}\right),\end{array}$$

where $\mathfrak{S}={\displaystyle \frac{\sqrt[n]{2S_c}}{2}}.$

(ii) If n is odd, then there is unique solution such that

$$\begin{array}{c}\hfill \mathfrak{S}\left(\begin{array}{cccccccc}1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3{u}_7}{S_c}}\\ {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_2{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{S_c}}\\ {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{S_c}}\\ {\displaystyle \frac{u_3}{S_c}}& -{\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& {\displaystyle \frac{{\mu }_3{u}_7}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_3{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_3{u}_4}{S_c}}\\ {\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{{\mu }_1{\mu }_2{u}_7}{S_c}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{S_c}}\\ {\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_2{u}_7}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_6}{S_c}}& {\displaystyle \frac{u_1}{S_c}}& 1& -{\displaystyle \frac{{\mu }_2{u}_3}{S_c}}& {\displaystyle \frac{{\mu }_2{u}_2}{S_c}}\\ {\displaystyle \frac{u_6}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_7}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& -{\displaystyle \frac{{\mu }_1{u}_5}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& {\displaystyle \frac{{\mu }_1{u}_3}{S_c}}& 1& -{\displaystyle \frac{{\mu }_1{u}_1}{S_c}}\\ {\displaystyle \frac{u_7}{S_c}}& {\displaystyle \frac{u_6}{S_c}}& -{\displaystyle \frac{u_5}{S_c}}& -{\displaystyle \frac{u_4}{S_c}}& {\displaystyle \frac{u_3}{S_c}}& {\displaystyle \frac{u_2}{S_c}}& -{\displaystyle \frac{u_1}{S_c}}& 1\end{array}\right),\end{array}$$

where $\mathfrak{S}={\displaystyle \frac{\sqrt[n]{2S_c}}{2}}.$

Proof. 

The proof can be shown using a method analogous to the one used in Theorem 13.    □

Theorem 17. 

Let ${\Phi }^{+}\left(c\right)$ be the right representation matrix of the generalized octonion c which has zero inner product and $\overrightarrow{u}$ be a unit lightlike vector with length 7. Then, the equation $x^n-c=0$ or $X^n-{\Phi }^{+}\left(c\right)=\mathbf{0}$ has the solutions in the following forms

(i) If n is even and $sgn\left(S_c\right)$ is positive, then there are two solutions such that

$$\begin{array}{c}\hfill X_0=\sqrt[n]{S_c}\left(\begin{array}{cccccccc}1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{n}}& {\displaystyle \frac{{\mu }_3{u}_4}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3{u}_7}{n}}\\ {\displaystyle \frac{u_1}{n}}& 1& {\displaystyle \frac{{\mu }_2{u}_3}{n}}& -{\displaystyle \frac{{\mu }_2{u}_2}{n}}& {\displaystyle \frac{{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}& {\displaystyle \frac{{\mu }_2{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{n}}\\ {\displaystyle \frac{u_2}{n}}& -{\displaystyle \frac{{\mu }_1{u}_3}{n}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_3{u}_6}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{n}}\\ {\displaystyle \frac{u_3}{n}}& -{\displaystyle \frac{u_2}{n}}& {\displaystyle \frac{u_1}{n}}& 1& {\displaystyle \frac{{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_3{u}_6}{n}}& {\displaystyle \frac{{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}\\ {\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_1{u}_5}{n}}& -{\displaystyle \frac{{\mu }_2{u}_6}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_2{u}_7}{n}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{n}}\\ {\displaystyle \frac{u_5}{n}}& -{\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_2{u}_7}{n}}& {\displaystyle \frac{{\mu }_2{u}_6}{n}}& {\displaystyle \frac{u_1}{n}}& 1& -{\displaystyle \frac{{\mu }_2{u}_3}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}\\ {\displaystyle \frac{u_6}{n}}& {\displaystyle \frac{{\mu }_1{u}_7}{n}}& -{\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_1{u}_5}{n}}& {\displaystyle \frac{u_2}{n}}& {\displaystyle \frac{{\mu }_1{u}_3}{n}}& 1& -{\displaystyle \frac{{\mu }_1{u}_1}{n}}\\ {\displaystyle \frac{u_7}{n}}& {\displaystyle \frac{u_6}{n}}& -{\displaystyle \frac{u_5}{n}}& -{\displaystyle \frac{u_4}{n}}& {\displaystyle \frac{u_3}{n}}& {\displaystyle \frac{u_2}{n}}& -{\displaystyle \frac{u_1}{n}}& 1\end{array}\right),\end{array}$$

$$\begin{array}{c}\hfill X_1=-\sqrt[n]{S_c}\left(\begin{array}{cccccccc}1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{n}}& {\displaystyle \frac{{\mu }_3{u}_4}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3{u}_7}{n}}\\ {\displaystyle \frac{u_1}{n}}& 1& {\displaystyle \frac{{\mu }_2{u}_3}{n}}& -{\displaystyle \frac{{\mu }_2{u}_2}{n}}& {\displaystyle \frac{{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}& {\displaystyle \frac{{\mu }_2{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{n}}\\ {\displaystyle \frac{u_2}{n}}& -{\displaystyle \frac{{\mu }_1{u}_3}{n}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_3{u}_6}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{n}}\\ {\displaystyle \frac{u_3}{n}}& -{\displaystyle \frac{u_2}{n}}& {\displaystyle \frac{u_1}{n}}& 1& {\displaystyle \frac{{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_3{u}_6}{n}}& {\displaystyle \frac{{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}\\ {\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_1{u}_5}{n}}& -{\displaystyle \frac{{\mu }_2{u}_6}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_2{u}_7}{n}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{n}}\\ {\displaystyle \frac{u_5}{n}}& -{\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_2{u}_7}{n}}& {\displaystyle \frac{{\mu }_2{u}_6}{n}}& {\displaystyle \frac{u_1}{n}}& 1& -{\displaystyle \frac{{\mu }_2{u}_3}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}\\ {\displaystyle \frac{u_6}{n}}& {\displaystyle \frac{{\mu }_1{u}_7}{n}}& -{\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_1{u}_5}{n}}& {\displaystyle \frac{u_2}{n}}& {\displaystyle \frac{{\mu }_1{u}_3}{n}}& 1& -{\displaystyle \frac{{\mu }_1{u}_1}{n}}\\ {\displaystyle \frac{u_7}{n}}& {\displaystyle \frac{u_6}{n}}& -{\displaystyle \frac{u_5}{n}}& -{\displaystyle \frac{u_4}{n}}& {\displaystyle \frac{u_3}{n}}& {\displaystyle \frac{u_2}{n}}& -{\displaystyle \frac{u_1}{n}}& 1\end{array}\right).\end{array}$$

Note that if $sgn\left(S_c\right)$ is negative, then there is no solution.

(ii) If n is odd, then there is unique solution such that

$$\begin{array}{c}\hfill \sqrt[n]{S_c}\left(\begin{array}{cccccccc}1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{n}}& {\displaystyle \frac{{\mu }_3{u}_4}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_2{\mu }_3{u}_7}{n}}\\ {\displaystyle \frac{u_1}{n}}& 1& {\displaystyle \frac{{\mu }_2{u}_3}{n}}& -{\displaystyle \frac{{\mu }_2{u}_2}{n}}& {\displaystyle \frac{{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}& {\displaystyle \frac{{\mu }_2{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_2{\mu }_3{u}_6}{n}}\\ {\displaystyle \frac{u_2}{n}}& -{\displaystyle \frac{{\mu }_1{u}_3}{n}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_3{u}_6}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_3{u}_5}{n}}\\ {\displaystyle \frac{u_3}{n}}& -{\displaystyle \frac{u_2}{n}}& {\displaystyle \frac{u_1}{n}}& 1& {\displaystyle \frac{{\mu }_3{u}_7}{n}}& -{\displaystyle \frac{{\mu }_3{u}_6}{n}}& {\displaystyle \frac{{\mu }_3{u}_5}{n}}& -{\displaystyle \frac{{\mu }_3{u}_4}{n}}\\ {\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_1{u}_5}{n}}& -{\displaystyle \frac{{\mu }_2{u}_6}{n}}& {\displaystyle \frac{{\mu }_1{\mu }_2{u}_7}{n}}& 1& {\displaystyle \frac{{\mu }_1{u}_1}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}& -{\displaystyle \frac{{\mu }_1{\mu }_2{u}_3}{n}}\\ {\displaystyle \frac{u_5}{n}}& -{\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_2{u}_7}{n}}& {\displaystyle \frac{{\mu }_2{u}_6}{n}}& {\displaystyle \frac{u_1}{n}}& 1& -{\displaystyle \frac{{\mu }_2{u}_3}{n}}& {\displaystyle \frac{{\mu }_2{u}_2}{n}}\\ {\displaystyle \frac{u_6}{n}}& {\displaystyle \frac{{\mu }_1{u}_7}{n}}& -{\displaystyle \frac{u_4}{n}}& -{\displaystyle \frac{{\mu }_1{u}_5}{n}}& {\displaystyle \frac{u_2}{n}}& {\displaystyle \frac{{\mu }_1{u}_3}{n}}& 1& -{\displaystyle \frac{{\mu }_1{u}_1}{n}}\\ {\displaystyle \frac{u_7}{n}}& {\displaystyle \frac{u_6}{n}}& -{\displaystyle \frac{u_5}{n}}& -{\displaystyle \frac{u_4}{n}}& {\displaystyle \frac{u_3}{n}}& {\displaystyle \frac{u_2}{n}}& -{\displaystyle \frac{u_1}{n}}& 1\end{array}\right).\end{array}$$

Proof. 

The proof can be given similar to Theorem 13.    □

6. An Application

In this section, we define the Matlab functions for calculations when we consider the Generalized Octonion c is in the real vector representation given in Theorem 11. Let us get $c_1=1+e_1+e_3+e_6+e_7,c_2=e_1+e_2+e_5+e_7\in \mathfrak{C}$ and ${\mu }_1,{\mu }_2,{\mu }_3$ be any real numbers. Then,

• The product of $c_1$ and $c_2$ is

  $$\begin{array}{c}\hfill c_1\times c_2={\mu }_1+{\mu }_1{\mu }_2{\mu }_3+(1+{\mu }_2-{\mu }_2{\mu }_3)e_1+(1-{\mu }_1-{\mu }_3)e_2\\ \hfill +(1-{\mu }_3)e_3+({\mu }_1-{\mu }_1{\mu }_2-{\mu }_2)e_4+(1-{\mu }_2)e_5+{\mu }_1{e}_6+2e_7\end{array}$$

  and by taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get $c_1\times c_2=1+e_1-2e_2-e_3+e_4+e_5+e_6+2e_7.$
• The inner product of $c_1$ and $c_2$ is $h(c_1,c_2)=-{\mu }_1(1+{\mu }_2{\mu }_3)$ and by taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get $h(c_1,c_2)=-1.$
• The norm of $c_1$ is $\left|\right|c_1\left|\right|=\sqrt{1-{\mu }_1+{\mu }_1{\mu }_2+{\mu }_2{\mu }_3-{\mu }_1{\mu }_2{\mu }_3}$ and the norm of $c_2$ is $\left|\right|c_2\left|\right|=\sqrt{-{\mu }_1-{\mu }_2+{\mu }_1{\mu }_3-{\mu }_1{\mu }_2{\mu }_3}.$ By taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get $\left|\right|c_1\left|\right|=0,\left|\right|c_2\left|\right|=1.$
• The inverse of $c_1$ does not exist and the inverse of $c_2$ is

  $$c_2^{-1}={\displaystyle \frac{-e_1-e_2-e_5-e_7}{\sqrt{-{\mu }_1-{\mu }_2+{\mu }_1{\mu }_3-{\mu }_1{\mu }_2{\mu }_3}}}.$$
  By taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get $c_2^{-1}=-e_1-e_2-e_5-e_7=\overline{c_2}.$
• The inner product of the vector parts of $c_1,c_2$ is $\langle V_{c_1},V_{c_2}\rangle ={\mu }_1+{\mu }_1{\mu }_2{\mu }_3$ and by taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get $\langle V_{c_1},V_{c_2}\rangle =1.$
• The vector product of the vector parts of $c_1,c_2$ is $V_{c_1}▴V_{c_2}=({\mu }_2-{\mu }_2{\mu }_3)e_1-({\mu }_1+{\mu }_1{\mu }_3)e_2+(1-{\mu }_3)e_3+({\mu }_1-{\mu }_2-{\mu }_1{\mu }_2)e_4-{\mu }_3{e}_5+{\mu }_1{e}_6+e_7,$ and by taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get $V_{c_1}▴V_{c_2}=-3e_2-e_3+e_4+e_6+e_7.$
• The right representation of the Generalized Octonion $c_1$ is

  $$\begin{array}{c}\hfill {\Phi }^{+}\left(c_1\right)=\left(\begin{array}{cccccccc}1& {\mu }_1& 0& -{\mu }_1{\mu }_2& 0& 0& -{\mu }_2{\mu }_3& {\mu }_1{\mu }_2{\mu }_3\\ 1& 1& {\mu }_2& 0& 0& 0& {\mu }_2{\mu }_3& -{\mu }_2{\mu }_3\\ 0& -{\mu }_1& 1& {\mu }_1& {\mu }_3& -{\mu }_1{\mu }_3& 0& 0\\ 1& 0& 1& 1& {\mu }_3& -{\mu }_3& 0& 0\\ 0& 0& -{\mu }_2& {\mu }_1{\mu }_2& 1& {\mu }_1& 0& -{\mu }_1{\mu }_2\\ 0& 0& -{\mu }_2& {\mu }_2& 1& 1& -{\mu }_2& 0\\ 1& {\mu }_1& 0& 0& 0& {\mu }_1& 1& -{\mu }_1\\ 1& 1& 0& 0& 1& 0& 1& 1\end{array}\right).\end{array}$$

  and by taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get

  $$\begin{array}{c}\hfill {\Phi }^{+}\left(c_1\right)=\left(\begin{array}{cccccccc}1& 1& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0& 0\\ 0& -1& 1& 1& 2& -2& 0& 0\\ 1& 0& 1& 1& 2& -2& 0& 0\\ 0& 0& 0& 0& 1& 1& 0& 0\\ 0& 0& 0& 0& 1& 1& 0& 0\\ 1& 1& 0& 0& 0& 1& 1& -1\\ 1& 1& 0& 0& 1& 0& 1& 1\end{array}\right).\end{array}$$
• The left representation of the Generalized Octonion $c_1$ is

  $$\begin{array}{c}\hfill {\Phi }^{-}\left(c_1\right)=\left(\begin{array}{cccccccc}1& {\mu }_1& 0& -{\mu }_1{\mu }_2& 0& 0& -{\mu }_2{\mu }_3& {\mu }_1{\mu }_2{\mu }_3\\ 1& 1& -{\mu }_2& 0& 0& 0& -{\mu }_2{\mu }_3& {\mu }_2{\mu }_3\\ 0& {\mu }_1& 1& -{\mu }_1& -{\mu }_3& {\mu }_1{\mu }_3& 0& 0\\ 1& 0& 1& 1& -{\mu }_3& {\mu }_3& 0& 0\\ 0& 0& {\mu }_2& -{\mu }_1{\mu }_2& 1& -{\mu }_1& 0& {\mu }_1{\mu }_2\\ 0& 0& {\mu }_2& -{\mu }_2& -1& 1& {\mu }_2& 0\\ 1& {\mu }_1& 0& 0& 0& -{\mu }_1& 1& {\mu }_1\\ 1& -1& 0& 0& -1& 0& 1& 1\end{array}\right).\end{array}$$

  and by taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ we get

  $$\begin{array}{c}\hfill {\Phi }^{+}\left(c_1\right)=\left(\begin{array}{cccccccc}1& 1& 0& 0& 0& 0& 0& 0\\ 1& 1& 0& 0& 0& 0& 0& 0\\ 0& 1& 1& -1& -2& 2& 0& 0\\ 1& 0& 1& 1& -2& 2& 0& 0\\ 0& 0& 0& 0& 1& -1& 0& 0\\ 0& 0& 0& 0& -1& 1& 0& 0\\ 1& 1& 0& 0& 0& -1& 1& 1\\ 1& -1& 0& 0& -1& 0& 1& 1\end{array}\right).\end{array}$$
• $h(c_1,c_1)=1-({\mu }_1-{\mu }_1{\mu }_2-{\mu }_2{\mu }_3+{\mu }_1{\mu }_2{\mu }_3)$ and $h(c_2,c_2)=-({\mu }_1+{\mu }_2-{\mu }_1{\mu }_3+{\mu }_1{\mu }_2{\mu }_3).$ By taking ${\mu }_1=1,{\mu }_2=0,{\mu }_3=2,$ the inner product $h(c_1,c_1)=0,$ so $c_1$ is a lightlike Generalized Octonion. Thus, $c_1=1+\overrightarrow{u},$ where $\overrightarrow{u}=e_1+e_3+e_6+e_7.$ Note that ${\overrightarrow{u}}^2=1.$ On the other hand, the inner product $h(c_2,c_2)=1>0,$ so $c_2$ is a spacelike Generalized Octonion. Also, $V_{c_2}^2=-1<0,$ so $c_2$ is a Generalized Octonion with spacelike vector part. Thus, $c_2=cos\alpha +\overrightarrow{u}sin\alpha ,$ where $cos\alpha =0,sin\alpha =1,\overrightarrow{u}=e_1+e_2+e_5+e_7,\alpha ={\displaystyle \frac{\pi }{2}}.$
• The 9^th power of $c_2$ is $c_2^9=cos\left(9\alpha \right)+\overrightarrow{u}sin\left(9\alpha \right)=c_2$. The right representation of $c_2$ is acquired from Theorem 12 case (iii) as follows:

  $$\begin{array}{c}\hfill {\Phi }^{+}\left(c_2\right)=\left(\begin{array}{cccccccc}0& {\mu }_1& {\mu }_2& 0& 0& -{\mu }_1{\mu }_3& 0& {\mu }_1{\mu }_2{\mu }_3\\ 1& 0& 0& -{\mu }_2& {\mu }_3& 0& {\mu }_2{\mu }_3& 0\\ 1& 0& 0& {\mu }_1& 0& -{\mu }_1{\mu }_3& 0& {\mu }_1{\mu }_3\\ 0& -1& 1& 0& {\mu }_3& 0& {\mu }_3& 0\\ 0& -{\mu }_1& 0& {\mu }_1{\mu }_2& 0& {\mu }_1& {\mu }_2& 0\\ 1& 0& -{\mu }_2& 0& 1& 0& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\mu }_1& 1& 0& 0& -{\mu }_1\\ 1& 0& -1& 0& 0& 1& -1& 0\end{array}\right).\end{array}$$
  Then, the 9^th power of ${\Phi }^{+}\left(c_2\right)$ is

  $$\begin{array}{c}\hfill \left(\begin{array}{cccccccc}0& {\mu }_1& {\mu }_2& 0& 0& -{\mu }_1{\mu }_3& 0& {\mu }_1{\mu }_2{\mu }_3\\ 1& 0& 0& -{\mu }_2& {\mu }_3& 0& {\mu }_2{\mu }_3& 0\\ 1& 0& 0& {\mu }_1& 0& -{\mu }_1{\mu }_3& 0& {\mu }_1{\mu }_3\\ 0& -1& 1& 0& {\mu }_3& 0& {\mu }_3& 0\\ 0& -{\mu }_1& 0& {\mu }_1{\mu }_2& 0& {\mu }_1& {\mu }_2& 0\\ 1& 0& -{\mu }_2& 0& 1& 0& 0& {\mu }_2\\ 0& {\mu }_1& 0& -{\mu }_1& 1& 0& 0& -{\mu }_1\\ 1& 0& -1& 0& 0& 1& -1& 0\end{array}\right).\end{array}$$
• The 3-roots of the equation $x^3-c_2=0$ are follows:

  $$\begin{array}{ccc}\hfill x_0& =& cos\left({\displaystyle \frac{\alpha }{3}}\right)+\overrightarrow{u}sin\left({\displaystyle \frac{\alpha }{3}}\right)={\displaystyle \frac{\sqrt{3}}{2}}+{\displaystyle \frac{1}{2}}(e_1+e_2+e_5+e_7).\hfill \\ \hfill x_1& =& cos\left({\displaystyle \frac{\alpha +2\pi }{3}}\right)+\overrightarrow{u}sin\left({\displaystyle \frac{\alpha +2\pi }{3}}\right)=-{\displaystyle \frac{\sqrt{3}}{2}}+{\displaystyle \frac{1}{2}}(e_1+e_2+e_5+e_7).\hfill \\ \hfill x_2& =& cos\left({\displaystyle \frac{\alpha +4\pi }{3}}\right)+\overrightarrow{u}sin\left({\displaystyle \frac{\alpha +4\pi }{3}}\right)=-(e_1+e_2+e_5+e_7).\hfill \end{array}$$

7. Conclusions

The purpose of this research is to examine some properties of Generalized Octonion which contain real octonions, split octonions, semi octonions, split semi octonions, quasi octonions, split quasi octonions and para octonions in special cases. The

Table 5. Research articles in the literature on special cases of ${\mu }_1,{\mu }_2,{\mu }_3$.

${\mu }_1=-1,{\mu }_2=-1,{\mu }_3=-1$	Real octonions [5]	Powers [18] Roots [18] Matrix representations [14] De Moivre of Matrix Representations [22]
${\mu }_1=-1,{\mu }_2=-1,{\mu }_3=1$	Split octonions [6]	Powers [15,18] Roots [15,18] Matrix representations [23] De Moivre of Matrix Representations [15]
${\mu }_1=-1,{\mu }_2=-1,{\mu }_3=0$	Semi octonions [7]	Powers [24] Roots [24] Matrix representations [7] De Moivre of Matrix Representations [7]
${\mu }_1=1,{\mu }_2=1,{\mu }_3=0$	Split semi octonions [8]	Powers [8] Roots [8] Matrix representations De Moivre of Matrix Representations
${\mu }_1=-1,{\mu }_2=0,{\mu }_3=0$	Quasi octonions [9]	Powers [9] Roots [9] Matrix representations De Moivre of Matrix Representations
${\mu }_1=1,{\mu }_2=0,{\mu }_3=0$	Split quasi octonions [10]	Powers [10] Roots [10] Matrix representations De Moivre of Matrix Representations
${\mu }_1=0,{\mu }_2=0,{\mu }_3=0$	Para octonions [11]	Powers [11] Roots [11] Matrix representations De Moivre of Matrix Representations

summarizes some studies in the literature on special cases with respect to ${\mu }_1,{\mu }_2,{\mu }_3$.

The generalized octonions are given in polar forms and are classified with respect to their inner products and vector parts. Powers and roots are investigated for the Generalized Octonions. Additionally, the matrix representation of Generalized Octonions are given with the algebraic properties. This section also presents the computation of powers and roots in the matrix representation of the Generalized Octonions. Moreover, MATLAB R2023a is used as a calculation tool for all results throughout the article and the MATLAB functions are defined in the last section with a numerical example, together.