Method for Constructing a Commutative Algebra of Hypercomplex Numbers

Abstract

Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers; (3) a method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N; and (4) the possibility of constructing a normed commutative algebra of other N-dimensional hypercomplex numbers with division over the field of real numbers. The article also shows that when using specific forms of representation of unit vectors, the product of vectors has the property of commutativity. Normed commutative algebras of N-dimensional hypercomplex numbers can be widely used to solve many topical scientific problems in the field of theoretical physics for modeling force fields with various types of symmetry, in cryptography for developing a number of new cryptographic programs using hypercomplex number algebras with different values of dimension, and in many other areas of fundamental and applied sciences.

1. Introduction

After successfully applying the algebra of complex numbers and functions of complex variables to solve a number of complex theoretical problems, many of the famous mathematicians in the world spent a lot of effort on solving the problem of constructing, first of all, an algebra of three-dimensional variables, in order to later construct algebras of multidimensional variables. Therefore, the quaternions proposed by W.R. Hamilton in 1843 gave rise to the rapid development of vector algebra and a number of other important sections of modern mathematics, which are an effective basis for building the fundamental foundations of theories and research methods in many areas of science and technology [1,2,3,4,5,6,7,8,9]. For example, vector algebra is the main mathematical tool for solving fundamental problems in the fields of mechanics [2,3,6,7], the theory of force fields [4,5], and the theoretical foundations for the formation of charged particle flows [8,9] in analytical instrumentation. Algebras of quaternions, octonions, and other hypercomplex numbers enable effective mathematical modeling of various physical processes and distributions of different types of force fields in spaces with different types of symmetry. Therefore, they are widely used to solve various fundamental and applied scientific problems; for example, they are used to solve a number of complex specific problems in the fields of theoretical and applied physics, robotics, cryptography, and digital processing of multidimensional signals [10,11,12,13,14,15,16,17,18,19,20,21,22]. For example, works [10,11,12,13,14,15,16] can be used in applied physics and robotics in the study of complex motions in space; in works [17,18,19,20,21,22], methods for using hypercomplex numbers in solving problems of relativistic physics and encoding multidimensional signals are given.

An analysis of the content of scientific papers and books devoted to theoretical problems of the algebra of hypercomplex numbers demonstrates that the construction of a commutative algebra of quaternions with division over the field of real numbers is considered impossible. There is practically no doubt about the existing theoretical statement that, unlike real and complex numbers, quaternions and other hypercomplex numbers cannot have the commutativity property. Doubts about this problem are also eliminated by the fact that the well-known theorems of Frobenius and Hurwitz also imply the impossibility of constructing a commutative algebra of quaternions, octonions, and other hypercomplex numbers [23].

The possibility of constructing a commutative division algebra of quaternions and other hypercomplex numbers over the field of real numbers was first considered in a recently published paper [24]. This article analyzes the product of quaternions and provides a proof of the theorem that the commutativity of the quaternion algebra can be ensured by setting a set of sign coefficients of the angles between the radius vectors in the coordinate planes of the vector part of the quaternion space coordinate system.

In this paper, we continue the discussion of the problem considered in [24] and study in more detail the condition for ensuring the commutativity of the algebra of hypercomplex numbers. Let us start with general questions, then move on to the formulations and proofs of theorems on the method of constructing a commutative algebra of hypercomplex numbers.

2. Materials and Methods

The field of complex numbers, as it is known, forms a commutative division algebra over the field of real numbers. A well-known extension of the complex numbers that form a non-commutative division algebra is the quaternions, which have the form

$$q=x_0+i{x}_1+j{x}_2+k{x}_3,$$

where $x_0,x_1,x_2,x_3$ are real numbers; i, j, k are imaginary units that satisfy the following conditions

$$i^2=j^2=k^2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j.$$

By doubling the quaternions, eight-dimensional numbers were obtained, which, according to the number of variables, were called octonions (octaves). Complex numbers, quaternions and octonions, and other hypercomplex numbers, as noted above, are currently very widely used in various branches of fundamental and applied mathematics.

Until now, it is considered impossible to construct hypercomplex numbers with division over the field of real numbers by other methods, except for doubling complex numbers, quaternions, and so on. Moreover, only algebras of numbers with dimensions 1, 2, 4, and 8 are recognized as exceptional division algebras. In addition, it is believed that only real and complex numbers have the property of commutativity of a product. In the traditional theory of hypercomplex numbers, it is considered proven that algebras of numbers with dimensions 4, 8, and so on cannot be commutative.

In the general case, in the canonical form, an $N$-dimensional hypercomplex number $h_N\left(x\right)$ can be written as

$$h_N\left(x\right)=x_0+{\displaystyle \sum _{n=1}^{N-1}{x}_n{i}_n},$$

(1)

where $x_0$ is a scalar component, variables $x_n$ ($n=1,2,\dots ,N-1$) make up the vector part of a hypercomplex number, and $i_n$ are imaginary units that satisfy the condition $i_n^2=-1$.

We start by considering the theorem on the commutativity of the product of quaternions. Taking into account (1), we write the quaternion $q=h_4\left(x\right)$ in the canonical form

$$q=s+i_{x}x+i_{y}y+i_{z}z,$$

(2)

where $s,x,y,z$ are real numbers and $i_x,i_y,i_z$ are imaginary units that satisfy the conditions $i_x^2=i_y^2=i_z^2=-1$.

Quaternions, when performing mathematical operations, will be further presented in a specific form

$$q_m=s_m+i_x^m{x}_m+i_y^m{y}_m+i_z^m{z}_m,$$

(3)

where the index m indicates that the quantities belong to a specific value of the quaternion. Note that when performing addition and subtraction operations for $m=a,b$ and in the general case

$$i_x^a=i_x^b=i_x, i_y^a=i_y^b=i_y, i_z^a=i_z^b=i_z.$$

Theorem 1. 

The commutativity of the product of quaternions $q_a=s_a+i_x^a{x}_a+i_y^a{y}_a+i_z^a{z}_a$ and $q_b=s_b+i_x^b{x}_b+i_y^b{y}_b+i_z^b{z}_b$ can be ensured under the following conditions $i_x^a{i}_x^b=i_y^a{i}_y^b=i_z^a{i}_z^b=-1$, $i_x^a{i}_y^b=-i_y^a{i}_x^b=-i_x^b{i}_y^a=i_y^b{i}_x^a=i_z$, $i_y^a{i}_z^b=-i_z^a{i}_y^b=-i_y^b{i}_z^a=i_z^b{i}_y^a=i_x$, and $i_z^a{i}_x^b=-i_x^a{i}_z^b=-i_z^b{i}_x^a=i_x^b{i}_z^a=i_y$.

Proof. 

First, let us take a closer look at the result of the multiplication $q_a{q}_b$.

$$\begin{array}{l}{q}_a{q}_b=\left(s_a+i_x^a{x}_a+i_y^a{y}_a+i_z^a{z}_a\right)\left(s_b+i_x^b{x}_b+i_y^b{y}_b+i_z^b{z}_b\right)\\ =s_a{s}_b-x_a{x}_b-y_a{y}_b-z_a{z}_b+s_a\left(i_x^b{x}_b+i_y^b{y}_b+i_z^b{z}_b\right)+s_b\left(i_x^a{x}_a+i_y^a{y}_a+i_z^a{z}_a\right)\\ +i_x^a{i}_y^b{x}_a{y}_b+i_y^a{i}_x^b{y}_a{x}_b+i_y^a{i}_z^b{y}_a{z}_b+i_z^a{i}_y^b{z}_a{y}_b+i_z^a{i}_x^b{z}_a{x}_b+i_x^a{i}_z^b{x}_a{z}_b\\ =s_a{s}_b-x_a{x}_b-y_a{y}_b-z_a{z}_b+i_x\left(s_a{x}_b+s_b{x}_a\right)+i_y\left(s_a{y}_b+s_b{y}_a\right)+i_z\left(s_a{z}_b+s_b{z}_a\right)\\ +i_x^a{i}_y^b\left(x_a{y}_b-y_a{x}_b\right)+i_y^a{i}_z^b\left(y_a{z}_b-z_a{y}_b\right)+i_z^a{i}_x^b\left(z_a{x}_b-x_a{z}_b\right)\\ =s_a{s}_b-x_a{x}_b-y_a{y}_b-z_a{z}_b+i_x\left(s_a{x}_b+s_b{x}_a\right)+i_y\left(s_a{y}_b+s_b{y}_a\right)+i_z\left(s_a{z}_b+s_b{z}_a\right)\\ +i_z\left(x_a{y}_b-y_a{x}_b\right)+i_x\left(y_a{z}_b-z_a{y}_b\right)+i_y\left(z_a{x}_b-x_a{z}_b\right).\end{array}$$

(4)

Now, let us swap the factors and also consider the result in detail $q_b{q}_a$.

$$\begin{array}{l}{q}_b{q}_a=\left(s_b+i_x^b{x}_b+i_y^b{y}_b+i_z^b{z}_b\right)\left(s_a+i_x^a{x}_a+i_y^a{y}_a+i_z^a{z}_a\right)\\ =s_b{s}_a-x_b{x}_a-y_b{y}_a-z_b{z}_a+s_b\left(i_x^a{x}_a+i_y^a{y}_a+i_z^a{z}_a\right)+s_a\left(i_x^b{x}_b+i_y^b{y}_b+i_z^b{z}_b\right)\\ +i_x^b{i}_y^a{x}_b{y}_a+i_y^b{i}_x^a{y}_b{x}_a+i_y^b{i}_z^a{y}_b{z}_a+i_z^b{i}_y^a{z}_b{y}_a+i_z^b{i}_x^a{z}_b{x}_a+i_x^b{i}_z^a{x}_b{z}_a\\ =s_b{s}_a-x_b{x}_a-y_b{y}_a-z_b{z}_a+i_x\left(s_b{x}_a+s_a{x}_b\right)+i_y\left(s_b{y}_a+s_a{y}_b\right)+i_z\left(s_b{z}_a+s_a{z}_b\right)\\ +i_y^b{i}_x^a\left(y_b{x}_a-x_b{y}_a\right)+i_z^b{i}_y^a\left(z_b{y}_a-y_b{z}_a\right)+i_x^b{i}_z^a\left(x_b{z}_a-z_b{x}_a\right)\\ =s_a{s}_b-x_a{x}_b-y_a{y}_b-z_a{z}_b+i_x\left(s_a{x}_b+s_b{x}_a\right)+i_y\left(s_a{y}_b+s_b{y}_a\right)+i_z\left(s_a{z}_b+s_b{z}_a\right)\\ +i_z\left(x_a{y}_b-y_a{x}_b\right)+i_x\left(y_a{z}_b-z_a{y}_b\right)+i_y\left(z_a{x}_b-x_a{z}_b\right).\end{array}$$

(5)

Comparison of the obtained results (4) and (5) confirms that the product of quaternions has the commutativity property

$$q_a{q}_b=q_b{q}_a.$$

Theorem 1 is proved. □

Remark 1. 

Taking $q_b={\overline{q}}_a=s_a-i_x^a{x}_a-i_y^a{y}_a-i_z^a{z}_a$ from (4), we get

$$q_a{\overline{q}}_a=s_a^2+x_a^2+y_a^2+z_a^2={\left|q_a\right|}^2.$$

(6)

Equation (6) implies that, under the considered method of ensuring the commutativity of the product, the quaternion algebra retains the property of a normed division algebra over the field of real numbers.

3. Results

Let us now consider, applying the method outlined above, the possibility of constructing a commutative algebra for other types of hypercomplex numbers. First, we formulate and prove a theorem on the commutativity of the product of octonions.

Theorem 2. 

The commutativity of the product of octonions $h_8^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^7{x}_n^a{i}_n^a}$ and $h_8^b\left(x\right)=x_0^b+{\displaystyle \sum _{n=1}^7{x}_n^b{i}_n^b}$ can be ensured if the rules for multiplying imaginary units are correctly specified and the following conditions $i_m^a{i}_m^b=i_n^a{i}_n^b=-1$ and $i_m^a{i}_n^b=-i_n^a{i}_m^b=-i_m^b{i}_n^a=i_n^b{i}_m^a=i_k$ are satisfied for $m\ne n$, where $m=1,2,\dots ,7$, and $k=1,2,\dots ,7$.

Proof. 

For octonions, the rules for multiplication of unit imaginary vectors indicated in Theorem 2 $i_1$–$i_7$ satisfy the conditions given in

Table 1. Multiplication table of imaginary units for the octonion.

1	$i_1^b$	$i_2^b$	$i_3^b$	$i_4^b$	$i_5^b$	$i_6^b$	$i_7^b$
$i_1^a$	−1	$i_3$	$-i_2$	$i_5$	$-i_4$	$-i_7$	$i_6$
$i_2^a$	$-i_3$	−1	$i_1$	$i_6$	$i_7$	$-i_4$	$-i_5$
$i_3^a$	$i_2$	$-i_1$	−1	$i_7$	$-i_6$	$i_5$	$-i_4$
$i_4^a$	$-i_5$	$-i_6$	$-i_7$	−1	$i_1$	$i_2$	$i_3$
$i_5^a$	$i_4$	$-i_7$	$i_6$	$-i_1$	−1	$-i_3$	$i_2$
$i_6^a$	$i_7$	$i_4$	$-i_5$	$-i_2$	$i_3$	−1	$-i_1$
$i_7^a$	$-i_6$	$i_5$	$i_4$	$-i_3$	$-i_2$	$i_1$	−1

.

Table 1 differs from the traditional table of octonions in that the imaginary units of the multiplier contain additional indices that indicate that each of the imaginary units belongs to a certain multiplier. Multiplying the octonions $h_8^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^7{x}_n^a{i}_n^a}$ and $h_8^b\left(x\right)=x_0^b+{\displaystyle \sum _{n=1}^7{x}_n^b{i}_n^b}$ using Table 1, we get

$$\begin{array}{l}{h}_8^a\left(x\right)\cdot h_8^b\left(x\right)=\left(x_0^a+x_1^a{i}_1^a+x_2^a{i}_2^a+x_3^a{i}_3^a+x_4^a{i}_4^a+x_5^a{i}_5^a+x_6^a{i}_6^a+x_7^a{i}_7^a\right)\\ \cdot \left(x_0^b+x_1^b{i}_1^b+x_2^b{i}_2^b+x_3^b{i}_3^b+x_4^b{i}_4^b+x_5^b{i}_5^b+x_6^b{i}_6^b+x_7^b{i}_7^b\right)\\ =x_0^a{x}_0^b-x_1^a{x}_1^b-x_2^a{x}_2^b-x_3^a{x}_3^b-x_4^a{x}_4^b-x_5^a{x}_5^b-x_6^a{x}_6^b-x_7^a{x}_7^b\\ +\left(x_0^a{x}_1^b+x_0^b{x}_1^a+x_2^a{x}_3^b-x_2^b{x}_3^a+x_4^a{x}_5^b-x_4^b{x}_5^a+x_7^a{x}_6^b-x_7^b{x}_6^a\right)i_1\\ +\left(x_0^a{x}_2^b+x_0^b{x}_2^a+x_3^a{x}_1^b-x_3^b{x}_1^a+x_4^a{x}_6^b-x_4^b{x}_6^a+x_5^a{x}_7^b-x_5^b{x}_7^a\right)i_2\\ +\left(x_0^a{x}_3^b+x_0^b{x}_3^a+x_1^a{x}_2^b-x_1^b{x}_2^a+x_6^a{x}_5^b-x_6^b{x}_5^a+x_4^a{x}_7^b-x_4^b{x}_7^a\right)i_3\\ +\left(x_0^a{x}_4^b+x_0^b{x}_4^a+x_5^a{x}_1^b-x_5^b{x}_1^a+x_6^a{x}_2^b-x_6^b{x}_2^a+x_7^a{x}_3^b-x_7^b{x}_3^a\right)i_4\\ +\left(x_0^a{x}_5^b+x_0^b{x}_5^a+x_1^a{x}_4^b-x_1^b{x}_4^a+x_3^a{x}_6^b-x_3^b{x}_6^a+x_7^a{x}_2^b-x_7^b{x}_2^a\right)i_5\\ +\left(x_0^a{x}_6^b+x_0^b{x}_6^a+x_1^a{x}_7^b-x_1^b{x}_7^a+x_2^a{x}_4^b-x_2^b{x}_4^a+x_5^a{x}_3^b-x_5^b{x}_3^a\right)i_6\\ +\left(x_0^a{x}_7^b+x_0^b{x}_7^a+x_6^a{x}_1^b-x_6^b{x}_1^a+x_2^a{x}_5^b-x_2^b{x}_5^a+x_3^a{x}_4^b-x_3^b{x}_4^a\right)i_7.\end{array}$$

(7)

Swapping the multipliers and performing the multiplication again, we can see that the product of octonions has the property of commutativity, that is

$$h_8^a\left(x\right)\cdot h_8^b\left(x\right)=h_8^b\left(x\right)\cdot h_8^a\left(x\right).$$

The conjugate number of the octonion $h_8^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^7{x}_n^a{i}_n^a}$ has the form ${\overline{h}}_8^a\left(x\right)=x_0^a-{\displaystyle \sum _{n=1}^7{x}_n^a{i}_n^a}$. Multiplying the octonion by its conjugate, we have

$$h_8^a\left(x\right)\cdot {\overline{h}}_8^b\left(x\right)={\left|h_8^a\left(x\right)\right|}^2={\displaystyle \sum _{n=0}^7{\left(x_n^a\right)}^2}.$$

(8)

Equation (8) shows that when performing the multiplication operation using Table 1, the octonion algebra has the property of a normed division algebra over the field of real numbers. Theorem 2 is proved. □

Remark 2. 

Commutative algebras of eight-dimensional hypercomplex numbers can also be constructed by using another multiplication table instead of Table 1, in which the conditions of Theorem 2 are satisfied. As an example, consider the possibility of constructing a commutative algebra using 

Table 2. Multiplication table of imaginary units for eight-dimensional hypercomplex numbers.

1	$i_1^b$	$i_2^b$	$i_3^b$	$i_4^b$	$i_5^b$	$i_6^b$	$i_7^b$
$i_1^a$	−1	$i_3$	$i_4$	$i_5$	$i_6$	$i_7$	$i_2$
$i_2^a$	$-i_3$	−1	$i_5$	$i_6$	$i_7$	$i_1$	$i_4$
$i_3^a$	$-i_4$	$-i_5$	−1	$i_7$	$i_1$	$i_2$	$i_6$
$i_4^a$	$-i_5$	$-i_6$	$-i_7$	−1	$i_2$	$i_3$	$i_1$
$i_5^a$	$-i_6$	$-i_7$	$-i_1$	$-i_2$	−1	$i_4$	$i_3$
$i_6^a$	$-i_7$	$-i_1$	$-i_2$	$-i_3$	$-i_4$	−1	$i_5$
$i_7^a$	$-i_2$	$-i_4$	$-i_6$	$-i_1$	$-i_3$	$-i_5$	−1

.

Let us now perform the multiplication of eight-dimensional hypercomplex numbers built on the basis of Table 2.

$$\begin{array}{l}{h}_8^a\left(x\right)\cdot h_8^b\left(x\right)=\left(x_0^a+x_1^a{i}_1^a+x_2^a{i}_2^a+x_3^a{i}_3^a+x_4^a{i}_4^a+x_5^a{i}_5^a+x_6^a{i}_6^a+x_7^a{i}_7^a\right)\\ \cdot \left(x_0^b+x_1^b{i}_1^b+x_2^b{i}_2^b+x_3^b{i}_3^b+x_4^b{i}_4^b+x_5^b{i}_5^b+x_6^b{i}_6^b+x_7^b{i}_7^b\right)\\ =x_0^a{x}_0^b-x_1^a{x}_1^b-x_2^a{x}_2^b-x_3^a{x}_3^b-x_4^a{x}_4^b-x_5^a{x}_5^b-x_6^a{x}_6^b-x_7^a{x}_7^b\\ +\left(x_0^a{x}_1^b+x_0^b{x}_1^a+x_2^a{x}_6^b-x_2^b{x}_6^a+x_3^a{x}_5^b-x_3^b{x}_5^a+x_4^a{x}_7^b-x_4^b{x}_7^a\right)i_1\\ +\left(x_0^a{x}_2^b+x_0^b{x}_2^a+x_1^a{x}_7^b-x_1^b{x}_7^a+x_4^a{x}_5^b-x_4^b{x}_5^a+x_3^a{x}_6^b-x_3^b{x}_6^a\right)i_2\\ +\left(x_0^a{x}_3^b+x_0^b{x}_3^a+x_1^a{x}_2^b-x_1^b{x}_2^a+x_4^a{x}_6^b-x_4^b{x}_6^a+x_5^a{x}_7^b-x_5^b{x}_7^a\right)i_3\\ +\left(x_0^a{x}_4^b+x_0^b{x}_4^a+x_1^a{x}_3^b-x_1^b{x}_3^a+x_5^a{x}_6^b-x_5^b{x}_6^a+x_2^a{x}_7^b-x_2^b{x}_7^a\right)i_4\\ +\left(x_0^a{x}_5^b+x_0^b{x}_5^a+x_1^a{x}_4^b-x_1^b{x}_4^a+x_2^a{x}_3^b-x_2^b{x}_3^a+x_6^a{x}_7^b-x_6^b{x}_7^a\right)i_5\\ +\left(x_0^a{x}_6^b+x_0^b{x}_6^a+x_1^a{x}_5^b-x_1^b{x}_5^a+x_2^a{x}_4^b-x_2^b{x}_4^a+x_3^a{x}_7^b-x_3^b{x}_7^a\right)i_6\\ +\left(x_0^a{x}_7^b+x_0^b{x}_7^a+x_1^a{x}_6^b-x_1^b{x}_6^a+x_2^a{x}_5^b-x_2^b{x}_5^a+x_3^a{x}_4^b-x_3^b{x}_4^a\right)i_7.\end{array}$$

(9)

From Equation (9) and the conditions of Theorem 2, it is easy to see that the eight-dimensional hypercomplex numbers constructed on the basis of Table 2 have the product commutativity property. The algebra of these hypercomplex numbers is also a normed algebra.

Remark 3. 

When compiling Table 2, the traditional rules for doubling quaternions were not used to construct eight-dimensional numbers. Therefore, we can conclude that the dimension of hypercomplex numbers with commutative algebra may not satisfy the condition $2^n$, where n = 1, 2, 3,…, N.

To confirm this conclusion, consider the possibility of constructing six-dimensional hypercomplex numbers based on

Table 3. Multiplication table of imaginary units for six-dimensional hypercomplex numbers.

1	$i_1^b$	$i_2^b$	$i_3^b$	$i_4^b$	$i_5^b$
$i_1^a$	−1	$i_3$	$-i_5$	$-i_2$	$i_4$
$i_2^a$	$-i_3$	−1	$i_4$	$-i_5$	$i_1$
$i_3^a$	$i_5$	$-i_4$	−1	$i_1$	$-i_2$
$i_4^a$	$i_2$	$i_5$	$-i_1$	−1	$-i_3$
$i_5^a$	$-i_4$	$-i_1$	$i_2$	$i_3$	−1

.

Multiplying the six-dimensional hypercomplex numbers $h_6^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^5{x}_n^a{i}_n^a}$ and $h_6^b\left(x\right)=x_0^b+{\displaystyle \sum _{n=1}^5{x}_n^b{i}_n^b}$ built on the basis of Table 3, we obtain

$$\begin{array}{l}{h}_6^a\left(x\right)\cdot h_6^b\left(x\right)=\left(x_0^a+x_1^a{i}_1^a+x_2^a{i}_2^a+x_3^a{i}_3^a+x_4^a{i}_4^a+x_5^a{i}_5^a\right)\\ \cdot \left(x_0^b+x_1^b{i}_1^b+x_2^b{i}_2^b+x_3^b{i}_3^b+x_4^b{i}_4^b+x_5^b{i}_5^b\right)\\ =x_0^a{x}_0^b-x_1^a{x}_1^b-x_2^a{x}_2^b-x_3^a{x}_3^b-x_4^a{x}_4^b-x_5^a{x}_5^b\\ +\left(x_0^a{x}_1^b+x_0^b{x}_1^a+x_2^a{x}_5^b-x_2^b{x}_5^a+x_3^a{x}_4^b-x_3^b{x}_4^a\right)i_1\\ +\left(x_0^a{x}_2^b+x_0^b{x}_2^a+x_4^a{x}_1^b-x_4^b{x}_1^a+x_5^a{x}_3^b-x_5^b{x}_3^a\right)i_2\\ +\left(x_0^a{x}_3^b+x_0^b{x}_3^a+x_1^a{x}_2^b-x_1^b{x}_2^a+x_5^a{x}_4^b-x_5^b{x}_4^a\right)i_3\\ +\left(x_0^a{x}_4^b+x_0^b{x}_4^a+x_2^a{x}_3^b-x_2^b{x}_3^a+x_1^a{x}_5^b-x_1^b{x}_5^a\right)i_4\\ +\left(x_0^a{x}_5^b+x_0^b{x}_5^a+x_3^a{x}_1^b-x_3^b{x}_1^a+x_4^a{x}_2^b-x_4^b{x}_2^a\right)i_5.\end{array}$$

(10)

From (10), it is easy to establish that the product of the six-dimensional numbers under consideration satisfies the commutativity properties of the product of factors.

According to Equation (1) above, an arbitrary six-dimensional number can be represented as

$$h_6\left(x\right)=x_0+x_1{i}_1+x_2{i}_2+x_3{i}_3+x_4{i}_4+x_5{i}_5.$$

(11)

Then, for the six-dimensional number (11), the conjugate number has the form

$${\overline{h}}_6\left(x\right)=x_0-x_1{i}_1-x_2{i}_2-x_3{i}_3-x_4{i}_4-x_5{i}_5.$$

(12)

From (10) to (12) it follows

$$h_6\left(x\right){\overline{h}}_6\left(x\right)=x_0^2+x_1^2+x_2^2+x_3^2+x_4^2+x_5^2.$$

(13)

Equation (13) indicates the possibility of constructing a normed division algebra for six-dimensional hypercomplex numbers over the field of real numbers.

The proof of the above theorems and the examples considered allow us to formulate the following lemma on the commutativity of the product of hypercomplex numbers with dimension $\mathrm{N}$.

Lemma 1. 

The commutativity of the product of hypercomplex numbers $h_N^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^{N-1}{x}_n^a{i}_n^a}$ and $h_N^b\left(x\right)=x_0^b+{\displaystyle \sum _{n=1}^{N-1}{x}_n^b{i}_n^b}$ with dimension $\mathrm{N}$ can be ensured if the rules for multiplying imaginary units are correctly specified and the following conditions, $i_m^a{i}_m^b=i_n^a{i}_n^b=-1$ and $i_m^a{i}_n^b=-i_n^a{i}_m^b=-i_m^b{i}_n^a=i_n^b{i}_m^a=i_k$, are met for $m\ne n\ne k$, where $m=1,2,\dots ,\mathrm{N}-1$, $n=1,2,\dots ,\mathrm{N}-1$ and $k=1,2,\dots ,\mathrm{N}-1$.

Proof. 

Let us analyze the above-specified multiplication tables that characterize hypercomplex numbers with specific dimension values. From all the tables considered above, it can be seen that the fulfillment of the conditions $i_m^a{i}_n^b=i_n^b{i}_m^a$ and $i_n^a{i}_m^b=i_m^b{i}_n^a$ is already laid down in the specified tables of hypercomplex numbers themselves. These conditions mean that the products of imaginary pseudovector units are commutative. In addition, the specified tables of hypercomplex numbers contain the conditions $i_m^a{i}_n^b=-i_m^b{i}_n^a=i_k$ and $i_n^b{i}_m^a=-i_n^a{i}_m^b=i_k$, which provide the possibility of constructing a normed division algebra over the field of real numbers. In the general case, the condition of the lemma $i_m^a{i}_n^b=-i_n^a{i}_m^b=-i_m^b{i}_n^a=i_n^b{i}_m^a=i_k$ contains the above necessary conditions for constructing a commutative and normed division algebra of hypercomplex numbers over the field of real numbers.

Conclusion: The application of the conditions of the lemma with well-defined general rules for multiplying imaginary units, in the form of tables or in another form, allows us to provide the necessary conditions for constructing a commutative algebra of hypercomplex numbers with dimension $\mathrm{N}$. The lemma is proved. □

We apply the above lemma to construct a set of concrete commutative algebras with dimension $\mathrm{N}$, where $\mathrm{N}$ is an even number.

Theorem 3. 

The commutativity of the product of hypercomplex numbers $h_N^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^{N-1}{x}_n^a{i}_n^a}$ and $h_N^b\left(x\right)=x_0^b+{\displaystyle \sum _{n=1}^{N-1}{x}_n^b{i}_n^b}$ with dimension $\mathrm{N}$, where $\mathrm{N}$ is an even number, can be ensured for $m\ne n\ne k$, where $m=1,2,\dots ,\mathrm{N}-1$, $n=1,2,\dots ,\mathrm{N}-1$ and $k=1,2,\dots ,\mathrm{N}-1$, with the following rules for multiplying imaginary units: $i_m^a{i}_m^b=i_n^a{i}_n^b=-1$; $i_m^a{i}_n^b=-i_n^a{i}_m^b=-i_m^b{i}_n^a=i_n^b{i}_m^a={\sigma }_k{i}_k$. Here, the index $k=\left(m+n\right)\mathrm{mod}\left(N-1\right)$ at $m=1,2,\dots ,\mathrm{N}-2$, $n=1,2,\dots ,\mathrm{N}-2$; $k=\left(m+m\right)\mathrm{mod}\left(N-1\right)$ at $m=\mathrm{N}-1$ and $k=\left(n+n\right)\mathrm{mod}\left(N-1\right)$ at $n=\mathrm{N}-1$; and value ${\sigma }_k=1$ if $m<n$ and ${\sigma }_k=-1$ if $m>n$.

Theorem 3 is a special case of the above lemma, so the proof of this theorem is the same as the proof of the lemma. Note that by changing the conditions for determining the sign coefficient ${\sigma }_k$, it is possible to generate hypercomplex numbers for modeling systems with different types of symmetry.

Let us give examples of the construction of hypercomplex numbers according to the rules given in this theorem. The first example is the eight-dimensional hypercomplex number discussed above with the multiplication rules shown in Table 2.

Let us take another example. For commutative ten-dimensional hypercomplex numbers, the multiplication table has the following form (

Table 4. Multiplication table of imaginary units for ten-dimensional hypercomplex numbers.

1	$i_1^b$	$i_2^b$	$i_3^b$	$i_4^b$	$i_5^b$	$i_6^b$	$i_7^b$	$i_8^b$	$i_9^b$
$i_1^a$	−1	$i_3$	$i_4$	$i_5$	$i_6$	$i_7$	$i_8$	$i_9$	$i_2$
$i_2^a$	$-i_3$	−1	$i_5$	$i_6$	$i_7$	$i_8$	$i_9$	$i_1$	$i_4$
$i_3^a$	$-i_4$	$-i_5$	−1	$i_7$	$i_8$	$i_9$	$i_1$	$i_2$	$i_6$
$i_4^a$	$-i_5$	$-i_6$	$-i_7$	−1	$i_9$	$i_1$	$i_2$	$i_3$	$i_8$
$i_5^a$	$-i_6$	$-i_7$	$-i_8$	$-i_9$	−1	$i_2$	$i_3$	$i_4$	$i_1$
$i_6^a$	$-i_7$	$-i_8$	$-i_9$	$-i_1$	$-i_2$	−1	$i_4$	$i_5$	$i_3$
$i_7^a$	$-i_8$	$-i_9$	$-i_1$	$-i_2$	$-i_3$	$-i_4$	−1	$i_6$	$i_5$
$i_8^a$	$-i_9$	$-i_1$	$-i_2$	$-i_3$	$-i_4$	$-i_5$	$-i_6$	−1	$i_7$
$i_9^a$	$-i_2$	$-i_4$	$-i_6$	$-i_8$	$-i_1$	$-i_3$	$-i_5$	$-i_7$	−1

):

The results of the multiplication of ten-dimensional hypercomplex numbers built on the basis of Table 4 are

$$\begin{array}{l}{h}_{10}^a\left(x\right)\cdot h_{10}^b\left(x\right)=\left(x_0^a+x_1^a{i}_1^a+x_2^a{i}_2^a+x_3^a{i}_3^a+x_4^a{i}_4^a+x_5^a{i}_5^a+x_6^a{i}_6^a+x_7^a{i}_7^a+x_8^a{i}_8^a+x_9^a{i}_9^a\right)\\ \cdot \left(x_0^b+x_1^b{i}_1^b+x_2^b{i}_2^b+x_3^b{i}_3^b+x_4^b{i}_4^b+x_5^b{i}_5^b+x_6^b{i}_6^b+x_7^b{i}_7^b+x_8^b{i}_8^b+x_9^b{i}_9^b\right)\\ =x_0^a{x}_0^b-x_1^a{x}_1^b-x_2^a{x}_2^b-x_3^a{x}_3^b-x_4^a{x}_4^b-x_5^a{x}_5^b-x_6^a{x}_6^b-x_7^a{x}_7^b-x_8^a{x}_8^b-x_9^a{x}_9^b\\ +\left(x_0^a{x}_1^b+x_0^b{x}_1^a+x_2^a{x}_8^b-x_2^b{x}_8^a+x_3^a{x}_7^b-x_3^b{x}_7^a+x_4^a{x}_6^b-x_4^b{x}_6^a+x_5^a{x}_9^b-x_5^b{x}_9^a\right)i_1\\ +\left(x_0^a{x}_2^b+x_0^b{x}_2^a+x_1^a{x}_9^b-x_1^b{x}_9^a+x_3^a{x}_8^b-x_3^b{x}_8^a+x_4^a{x}_7^b-x_4^b{x}_7^a+x_5^a{x}_6^b-x_5^b{x}_6^a\right)i_2\\ +\left(x_0^a{x}_3^b+x_0^b{x}_3^a+x_1^a{x}_2^b-x_1^b{x}_2^a+x_4^a{x}_8^b-x_4^b{x}_8^a+x_5^a{x}_7^b-x_5^b{x}_7^a+x_6^a{x}_9^b-x_6^b{x}_9^a\right)i_3\\ +\left(x_0^a{x}_4^b+x_0^b{x}_4^a+x_1^a{x}_3^b-x_1^b{x}_3^a+x_2^a{x}_9^b-x_2^b{x}_9^a+x_5^a{x}_8^b-x_5^b{x}_8^a+x_6^a{x}_7^b-x_6^b{x}_7^a\right)i_4\\ +\left(x_0^a{x}_5^b+x_0^b{x}_5^a+x_1^a{x}_4^b-x_1^b{x}_4^a+x_2^a{x}_3^b-x_2^b{x}_3^a+x_6^a{x}_8^b-x_6^b{x}_8^a+x_7^a{x}_9^b-x_7^b{x}_9^a\right)i_5\\ +\left(x_0^a{x}_6^b+x_0^b{x}_6^a+x_1^a{x}_5^b-x_1^b{x}_5^a+x_2^a{x}_4^b-x_2^b{x}_4^a+x_3^a{x}_9^b-x_3^b{x}_9^a+x_7^a{x}_8^b-x_7^b{x}_8^a\right)i_6\\ +\left(x_0^a{x}_7^b+x_0^b{x}_7^a+x_1^a{x}_6^b-x_1^b{x}_6^a+x_2^a{x}_5^b-x_2^b{x}_5^a+x_3^a{x}_4^b-x_3^b{x}_4^a+x_8^a{x}_9^b-x_8^b{x}_9^a\right)i_7\\ +\left(x_0^a{x}_8^b+x_0^b{x}_8^a+x_1^a{x}_7^b-x_1^b{x}_7^a+x_2^a{x}_6^b-x_2^b{x}_6^a+x_3^a{x}_5^b-x_3^b{x}_5^a+x_4^a{x}_9^b-x_4^b{x}_9^a\right)i_8\\ +\left(x_0^a{x}_9^b+x_0^b{x}_9^a+x_1^a{x}_8^b-x_1^b{x}_8^a+x_2^a{x}_7^b-x_2^b{x}_7^a+x_3^a{x}_6^b-x_3^b{x}_6^a+x_4^a{x}_5^b-x_4^b{x}_5^a\right)i_9.\end{array}$$

It is easy to see from this equation that the product of the considered ten-dimensional hypercomplex numbers satisfies the commutativity properties of the product of factors.

For a given ten-dimensional hypercomplex number $h_{10}^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^9{x}_n^a{i}_n^a}$, the conjugate number has the form ${\overline{h}}_{10}^a\left(x\right)=x_0^a-{\displaystyle \sum _{n=1}^9{x}_n^a{i}_n^a}$, and the modulus is determined by the formula

$${\left|h_{10}^a\left(x\right)\right|}^2=h_{10}^a\left(x\right)\cdot {\overline{h}}_{10}^a\left(x\right)={\displaystyle \sum _{n=0}^9{\left(x_n^a\right)}^2}.$$

It follows from this formula that when performing the multiplication operation using Table 4, the algebra of ten-dimensional hypercomplex numbers has the property of a normed division algebra over the field of real numbers.

On the basis of studies of the properties of normed algebras by Hurwitz and division algebras over the field of real numbers by Frobenius in the theory of hypercomplex numbers, a conclusion was made about identities for normed division algebras over the field of real numbers. These identities generally have the form

$${\left|uv\right|}^2={\left|u\right|}^2{\left|v\right|}^2.$$

Here, $u=h_N^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^{N-1}{x}_n^a{i}_n^a}$ and $v=h_N^b\left(x\right)=x_0^b+{\displaystyle \sum _{n=1}^{N-1}{x}_n^b{i}_n^b}$.

Algebras for which this identity holds were considered exceptional normed division algebras over the field of real numbers. Note that the theorems of Hurwitz and Frobenius were mainly concerned with the definition of properties of associative and alternative algebras of hypercomplex numbers. The method proposed in this paper for constructing commutative algebras of normed hypercomplex numbers significantly expands the list of algebras that satisfy the above identity. For example, the identity for the six-dimensional algebra considered above has the form

$$\begin{array}{l}{\left|h_6^a\left(x\right)\right|}^2{\left|h_6^b\left(x\right)\right|}^2={\left|h_6^a\left(x\right)h_6^b\left(x\right)\right|}^2\\ =\left({\left(x_0^a\right)}^2+{\left(x_1^a\right)}^2+{\left(x_2^a\right)}^2+{\left(x_3^a\right)}^2+{\left(x_4^a\right)}^2+{\left(x_5^a\right)}^2\right)\\ \cdot \left({\left(x_0^b\right)}^2+{\left(x_1^b\right)}^2+{\left(x_2^b\right)}^2+{\left(x_3^b\right)}^2+{\left(x_4^b\right)}^2+{\left(x_5^b\right)}^2\right)\\ ={\left(x_0^a{x}_0^b-x_1^a{x}_1^b-x_2^a{x}_2^b-x_3^a{x}_3^b-x_4^a{x}_4^b-x_5^a{x}_5^b\right)}^2\\ +{\left(x_0^a{x}_1^b+x_0^b{x}_1^a+x_2^a{x}_5^b-x_2^b{x}_5^a+x_3^a{x}_4^b-x_3^b{x}_4^a\right)}^2\\ +{\left(x_0^a{x}_2^b+x_0^b{x}_2^a+x_4^a{x}_1^b-x_4^b{x}_1^a+x_5^a{x}_3^b-x_5^b{x}_3^a\right)}^2\\ +{\left(x_0^a{x}_3^b+x_0^b{x}_3^a+x_1^a{x}_2^b-x_1^b{x}_2^a+x_5^a{x}_4^b-x_5^b{x}_4^a\right)}^2\\ +{\left(x_0^a{x}_4^b+x_0^b{x}_4^a+x_2^a{x}_3^b-x_2^b{x}_3^a+x_1^a{x}_5^b-x_1^b{x}_5^a\right)}^2\\ +{\left(x_0^a{x}_5^b+x_0^b{x}_5^a+x_3^a{x}_1^b-x_3^b{x}_1^a+x_4^a{x}_2^b-x_4^b{x}_2^a\right)}^2.\end{array}$$

A similar identity can be given for ten-dimensional hypercomplex numbers $h_{10}^a\left(x\right)=x_0^a+{\displaystyle \sum _{n=1}^9{x}_n^a{i}_n^a}$ и $h_{10}^b\left(x\right)=x_0^b+{\displaystyle \sum _{n=1}^9{x}_n^b{i}_n^b}$.

$$\begin{array}{l}{\left|h_{10}^a\left(x\right)\right|}^2{\left|h_{10}^b\left(x\right)\right|}^2={\left|h_{10}^a\left(x\right)h_{10}^b\left(x\right)\right|}^2\\ =\left({\left(x_0^a\right)}^2+{\left(x_1^a\right)}^2+{\left(x_2^a\right)}^2+{\left(x_3^a\right)}^2+{\left(x_4^a\right)}^2+{\left(x_5^a\right)}^2+{\left(x_6^a\right)}^2+{\left(x_7^a\right)}^2+{\left(x_8^a\right)}^2+{\left(x_9^a\right)}^2\right)\\ \cdot \left({\left(x_0^b\right)}^2+{\left(x_1^b\right)}^2+{\left(x_2^b\right)}^2+{\left(x_3^b\right)}^2+{\left(x_4^b\right)}^2+{\left(x_5^b\right)}^2+{\left(x_6^b\right)}^2+{\left(x_7^b\right)}^2+{\left(x_8^b\right)}^2+{\left(x_9^b\right)}^2\right)\\ ={\left(x_0^a{x}_0^b-x_1^a{x}_1^b-x_2^a{x}_2^b-x_3^a{x}_3^b-x_4^a{x}_4^b-x_5^a{x}_5^b-x_6^a{x}_6^b-x_7^a{x}_7^b-x_8^a{x}_8^b-x_9^a{x}_9^b\right)}^2\\ +{\left(x_0^a{x}_1^b+x_0^b{x}_1^a+x_2^a{x}_8^b-x_2^b{x}_8^a+x_3^a{x}_7^b-x_3^b{x}_7^a+x_4^a{x}_6^b-x_4^b{x}_6^a+x_5^a{x}_9^b-x_5^b{x}_9^a\right)}^2\\ +{\left(x_0^a{x}_2^b+x_0^b{x}_2^a+x_1^a{x}_9^b-x_1^b{x}_9^a+x_3^a{x}_8^b-x_3^b{x}_8^a+x_4^a{x}_7^b-x_4^b{x}_7^a+x_5^a{x}_6^b-x_5^b{x}_6^a\right)}^2\\ +{\left(x_0^a{x}_3^b+x_0^b{x}_3^a+x_1^a{x}_2^b-x_1^b{x}_2^a+x_4^a{x}_8^b-x_4^b{x}_8^a+x_5^a{x}_7^b-x_5^b{x}_7^a+x_6^a{x}_9^b-x_6^b{x}_9^a\right)}^2\\ +{\left(x_0^a{x}_4^b+x_0^b{x}_4^a+x_1^a{x}_3^b-x_1^b{x}_3^a+x_2^a{x}_9^b-x_2^b{x}_9^a+x_5^a{x}_8^b-x_5^b{x}_8^a+x_6^a{x}_7^b-x_6^b{x}_7^a\right)}^2\\ +{\left(x_0^a{x}_5^b+x_0^b{x}_5^a+x_1^a{x}_4^b-x_1^b{x}_4^a+x_2^a{x}_3^b-x_2^b{x}_3^a+x_6^a{x}_8^b-x_6^b{x}_8^a+x_7^a{x}_9^b-x_7^b{x}_9^a\right)}^2\\ +{\left(x_0^a{x}_6^b+x_0^b{x}_6^a+x_1^a{x}_5^b-x_1^b{x}_5^a+x_2^a{x}_4^b-x_2^b{x}_4^a+x_3^a{x}_9^b-x_3^b{x}_9^a+x_7^a{x}_8^b-x_7^b{x}_8^a\right)}^2\\ +{\left(x_0^a{x}_7^b+x_0^b{x}_7^a+x_1^a{x}_6^b-x_1^b{x}_6^a+x_2^a{x}_5^b-x_2^b{x}_5^a+x_3^a{x}_4^b-x_3^b{x}_4^a+x_8^a{x}_9^b-x_8^b{x}_9^a\right)}^2\\ +{\left(x_0^a{x}_8^b+x_0^b{x}_8^a+x_1^a{x}_7^b-x_1^b{x}_7^a+x_2^a{x}_6^b-x_2^b{x}_6^a+x_3^a{x}_5^b-x_3^b{x}_5^a+x_4^a{x}_9^b-x_4^b{x}_9^a\right)}^2\\ +{\left(x_0^a{x}_9^b+x_0^b{x}_9^a+x_1^a{x}_8^b-x_1^b{x}_8^a+x_2^a{x}_7^b-x_2^b{x}_7^a+x_3^a{x}_6^b-x_3^b{x}_6^a+x_4^a{x}_5^b-x_4^b{x}_5^a\right)}^2.\end{array}$$

As noted above, vector algebra developed rapidly after Hamilton’s proposal of the quaternion algebra. The space of vectors, as is known, can be considered as a part of the space of quaternions. Therefore, the method of ensuring the commutativity of the product of quaternions must also lead to the commutativity of the vector (orthogonal) product of vectors. In canonical form, a three-dimensional vector $\overrightarrow{r}$ has the form

$$\overrightarrow{r}=x{\overrightarrow{e}}_x+y{\overrightarrow{e}}_y+z{\overrightarrow{e}}_z,$$

(14)

where $x,y,z$ are the coordinates of the axes in the coordinate system of three-dimensional space, and ${\overrightarrow{e}}_x,{\overrightarrow{e}}_y,{\overrightarrow{e}}_z$ are the unit direction vectors in the Cartesian coordinate system.

Theorem 4. 

The commutativity of the vector (orthogonal) product of vectors ${\overrightarrow{r}}_a=x_a{\overrightarrow{e}}_x^a+y_a{\overrightarrow{e}}_y^a+z_a{\overrightarrow{e}}_z^a$ and ${\overrightarrow{r}}_b=x_b{\overrightarrow{e}}_x^b+y_b{\overrightarrow{e}}_y^b+z_b{\overrightarrow{e}}_z^b$ can be ensured under the following conditions: ${\overrightarrow{e}}_x^a{\overrightarrow{e}}_x^b={\overrightarrow{e}}_y^a{\overrightarrow{e}}_y^b={\overrightarrow{e}}_z^a{\overrightarrow{e}}_z^b=0$, ${\overrightarrow{e}}_x^a{\overrightarrow{e}}_y^b=-{\overrightarrow{e}}_y^a{\overrightarrow{e}}_x^b=-{\overrightarrow{e}}_x^b{\overrightarrow{e}}_y^a={\overrightarrow{e}}_y^b{\overrightarrow{e}}_x^a={\overrightarrow{e}}_z$, and ${\overrightarrow{e}}_y^a{\overrightarrow{e}}_z^b=-{\overrightarrow{e}}_z^a{\overrightarrow{e}}_y^b=-{\overrightarrow{e}}_y^b{\overrightarrow{e}}_z^a={\overrightarrow{e}}_z^b{\overrightarrow{e}}_y^a={\overrightarrow{e}}_x$, ${\overrightarrow{e}}_z^a{\overrightarrow{e}}_x^b=-{\overrightarrow{e}}_x^a{\overrightarrow{e}}_z^b=-{\overrightarrow{e}}_z^b{\overrightarrow{e}}_x^a={\overrightarrow{e}}_x^b{\overrightarrow{e}}_z^a={\overrightarrow{e}}_y$.

Proof. 

Let us perform the multiplication of vectors ${\overrightarrow{r}}_a=x_a{\overrightarrow{e}}_x^a+y_a{\overrightarrow{e}}_y^a+z_a{\overrightarrow{e}}_z^a$ and ${\overrightarrow{r}}_b=x_b{\overrightarrow{e}}_x^b+y_b{\overrightarrow{e}}_y^b+z_b{\overrightarrow{e}}_z^b$ in accordance with the conditions specified in Theorem 3.

$$\begin{array}{l}{\overrightarrow{r}}_a{\overrightarrow{r}}_b=\left(x_a{\overrightarrow{e}}_x^a+y_a{\overrightarrow{e}}_y^a+z_a{\overrightarrow{e}}_z^a\right)\left(x_b{\overrightarrow{e}}_x^b+y_b{\overrightarrow{e}}_y^b+z_b{\overrightarrow{e}}_z^b\right)\\ =x_a{y}_b{\overrightarrow{e}}_x^a{\overrightarrow{e}}_y^b+y_a{x}_b{\overrightarrow{e}}_y^a{\overrightarrow{e}}_x^b+y_a{z}_b{\overrightarrow{e}}_y^a{\overrightarrow{e}}_z^b+z_a{y}_b{\overrightarrow{e}}_z^a{\overrightarrow{e}}_y^b+z_a{x}_b{\overrightarrow{e}}_z^a{\overrightarrow{e}}_x^b+x_a{z}_b{\overrightarrow{e}}_x^a{\overrightarrow{e}}_z^b\\ ={\overrightarrow{e}}_x^a{\overrightarrow{e}}_y^b\left(x_a{y}_b-y_a{x}_b\right)+{\overrightarrow{e}}_y^a{\overrightarrow{e}}_z^b\left(y_a{z}_b-z_a{y}_b\right)+{\overrightarrow{e}}_z^a{\overrightarrow{e}}_x^b\left(z_a{x}_b-x_a{z}_b\right)\\ ={\overrightarrow{e}}_y^b{\overrightarrow{e}}_x^a\left(x_a{y}_b-y_a{x}_b\right)+{\overrightarrow{e}}_z^b{\overrightarrow{e}}_y^a\left(y_a{z}_b-z_a{y}_b\right)+{\overrightarrow{e}}_x^b{\overrightarrow{e}}_z^a\left(z_a{x}_b-x_a{z}_b\right)\\ =\left(x_a{y}_b-y_a{x}_b\right){\overrightarrow{e}}_z+\left(y_a{z}_b-z_a{y}_b\right){\overrightarrow{e}}_x+\left(z_a{x}_b-x_a{z}_b\right){\overrightarrow{e}}_y={\overrightarrow{r}}_b{\overrightarrow{r}}_a.\end{array}$$

(15)

Theorem 4 is proved. □

The method of constructing a commutative algebra of hypercomplex numbers considered in the article and the theorem on the commutativity of the vector (orthogonal) product of vectors can be used to solve a wide range of fundamental and applied scientific problems.

4. Discussion

It follows from the foregoing that the main condition for constructing a commutative algebra of hypercomplex numbers is the need to indicate that the imaginary units belong to one or another factor when performing the multiplication operation. Unfortunately, from the moment Hamilton proposed the foundations of quaternion algebra and vector algebra until now, no one paid attention to this condition.

To justify the need to denote the belonging of imaginary units or unit vectors to one or another factor when performing the operation of multiplying hypercomplex numbers or vectors, consider the following two-dimensional vector $\overrightarrow{R}=\overrightarrow{R}\left(x,y\right)$.

$$\overrightarrow{R}=x{\overrightarrow{e}}_x+y{\overrightarrow{e}}_y,$$

(16)

where ${\overrightarrow{e}}_x$ and ${\overrightarrow{e}}_y$ are the unit direction vectors, $x$ and $y$ denote the coordinate values of the point in the Cartesian coordinate system.

The same vector, with modulus $R$ and eigendirection $\overrightarrow{e}$, can also be represented as

$$\overrightarrow{R}=R\overrightarrow{e}.$$

(17)

From formulas (1) and (2) it follows

$$\overrightarrow{e}=\frac{\overrightarrow{R}}{R}=\frac{x}{R}{\overrightarrow{e}}_x+\frac{y}{R}{\overrightarrow{e}}_y={\overrightarrow{e}}_x\cos \phi +{\overrightarrow{e}}_y\sin \phi .$$

Here, the coordinate $\phi$ describes, in addition to the linear directions $x$ and $y$, the rotational direction.

The rotation of a coordinate system $x^{\prime }$, $y^{\prime }$ or individual vectors about an axis $x$ by an angle $\phi$ can be described using a unit rotation vector

$${\overrightarrow{e}}_{\phi }={\overrightarrow{e}}_x\cos {\phi }_{xy}+{\overrightarrow{e}}_y\sin {\phi }_{xy}.$$

(18)

To denote the direction of the angular coordinate in Equation (18), the angle $\phi$ is written with the index ${\phi }_{xy}$. The index indicates the direction of rotation of the unit vector around the origin of the planar coordinate system. In (18), the angle of rotation ${\phi }_{xy}=\left(\stackrel{\wedge }{{\overrightarrow{e}}_x,{\overrightarrow{e}}_{\phi }}\right)$ is measured from the axis $x$ in the direction of the axis $y$.

Let us assume that when the coordinate system $x^{\prime }$, $y^{\prime }$ is rotated, the unit vector (18) describes the angular direction of displacement of the abscissa of the rotating rectangular coordinate system relative to the initial coordinate system $x$, $y$. In this case, the direction of the ordinate of the rotating coordinate system is described by the unit vector ${\overrightarrow{e}}_{\phi }^{\perp }$

$${\overrightarrow{e}}_{\phi }^{\perp }=\overrightarrow{e}\left({\phi }_{xy}+\frac{\pi }{2}\right)=-{\overrightarrow{e}}_x\sin {\phi }_{xy}+{\overrightarrow{e}}_y\cos {\phi }_{xy}.$$

(19)

The modules of all the above unit vectors are equal to one. From (18) and (19) it follows

$${\overrightarrow{e}}_x={\overrightarrow{e}}_{\phi }\cos {\phi }_{xy}-{\overrightarrow{e}}_{\phi }^{\perp }\sin {\phi }_{xy},$$

(20)

$${\overrightarrow{e}}_y={\overrightarrow{e}}_{\phi }\sin {\phi }_{xy}+{\overrightarrow{e}}_{\phi }^{\perp }\cos {\phi }_{xy}.$$

(21)

Equations (20) and (21) determine the values of the unit vectors ${\overrightarrow{e}}_x$ and ${\overrightarrow{e}}_y$ with respect to the radial ${\overrightarrow{e}}_{\phi }={\overrightarrow{e}}_{\rho }$ and ${\overrightarrow{e}}_{\phi }^{\perp }={\overrightarrow{e}}_{\tau }$ orthogonal (tangential) unit vectors of the rotating coordinate system $x^{\prime }$, $y^{\prime }$. In principle, the coordinate system $x$, $y$, can be considered a rotating coordinate system with respect to the coordinate system $x^{\prime }$, $y^{\prime }$. Then, for a specific vector ${\overrightarrow{R}}_n=R_n{\overrightarrow{e}}_n$, you must specify specific values of the angle ${\phi }_{xy}^n$. In this case, Equations (20) and (21) take the form

$${\overrightarrow{e}}_x^n={\overrightarrow{e}}_x({\phi }_{xy}^n)={\overrightarrow{e}}_{\rho }\cos {\phi }_{xy}^n-{\overrightarrow{e}}_{\tau }\sin {\phi }_{xy}^n,$$

(22)

$${\overrightarrow{e}}_y^n={\overrightarrow{e}}_y({\phi }_{xy}^n)={\overrightarrow{e}}_{\rho }sin{\phi }_{xy}^n+{\overrightarrow{e}}_{\tau }\cos {\phi }_{xy}^n.$$

(23)

Equations (20)–(23) satisfy the conditions

$${\left({\overrightarrow{e}}_x\right)}^2={\left({\overrightarrow{e}}_y\right)}^2={\left({\overrightarrow{e}}_{\rho }\right)}^2={\left({\overrightarrow{e}}_{\tau }\right)}^2=1$$

and

$${\overrightarrow{e}}_x{\overrightarrow{e}}_y={\overrightarrow{e}}_{\rho }{\overrightarrow{e}}_{\tau }=0.$$

These canonical conditions are valid for a single vector or for a pair of collinear vectors.

Consider Equations (22) and (23) for a pair of collinear and noncollinear unit vectors. Under the conditions from (22) and (23) for collinear vectors, we obtain

$${\overrightarrow{e}}_x^a{\overrightarrow{e}}_x^b={\overrightarrow{e}}_y^a{\overrightarrow{e}}_y^b=1, {\overrightarrow{e}}_x^a{\overrightarrow{e}}_y^b=0.$$

(24)

Equation (24) reflects the rules of the scalar product.

For noncollinear unit vectors, we have

$$\begin{array}{l}{\overrightarrow{e}}_x^a{\overrightarrow{e}}_x^b=\left({\overrightarrow{e}}_{\rho }\cos {\phi }_{xy}^a-{\overrightarrow{e}}_{\tau }\sin {\phi }_{xy}^a\right)\left({\overrightarrow{e}}_{\rho }\cos {\phi }_{xy}^b-{\overrightarrow{e}}_{\tau }\sin {\phi }_{xy}^b\right)\\ =\cos {\phi }_{xy}^a\cos {\phi }_{xy}^b+\sin {\phi }_{xy}^a\sin {\phi }_{xy}^b=\cos \left({\phi }_{xy}^b-{\phi }_{xy}^a\right),\end{array}$$

(25)

$$\begin{array}{l}{\overrightarrow{e}}_y^a{\overrightarrow{e}}_y^b=\left({\overrightarrow{e}}_{\rho }\sin {\phi }_{xy}^a+{\overrightarrow{e}}_{\tau }\cos {\phi }_{xy}^a\right)\left({\overrightarrow{e}}_{\rho }\sin {\phi }_{xy}^b+{\overrightarrow{e}}_{\tau }\cos {\phi }_{xy}^b\right)\\ =\sin {\phi }_{xy}^a\sin {\phi }_{xy}^b+\cos {\phi }_{xy}^a\cos {\phi }_{xy}^b=\cos \left({\phi }_{xy}^b-{\phi }_{xy}^a\right),\end{array}$$

(26)

$$\begin{array}{l}{\overrightarrow{e}}_x^a{\overrightarrow{e}}_y^b=\left({\overrightarrow{e}}_{\rho }\cos {\phi }_{xy}^a-{\overrightarrow{e}}_{\tau }\sin {\phi }_{xy}^a\right)\left({\overrightarrow{e}}_{\rho }\sin {\phi }_{xy}^b+{\overrightarrow{e}}_{\tau }\cos {\phi }_{xy}^b\right)\\ =\cos {\phi }_{xy}^a\sin {\phi }_{xy}^b-\sin {\phi }_{xy}^a\cos {\phi }_{xy}^b=\sin \left({\phi }_{xy}^b-{\phi }_{xy}^a\right).\end{array}$$

(27)

From (25) to (27) it follows

$${\overrightarrow{e}}_x^a{\overrightarrow{e}}_x^b={\overrightarrow{e}}_x^b{\overrightarrow{e}}_x^a={\overrightarrow{e}}_y^b{\overrightarrow{e}}_y^a={\overrightarrow{e}}_y^a{\overrightarrow{e}}_y^b=\cos \left({\phi }_{xy}^b-{\phi }_{xy}^a\right),$$

(28)

$${\overrightarrow{e}}_x^a{\overrightarrow{e}}_y^b=-{\overrightarrow{e}}_x^b{\overrightarrow{e}}_y^a=-{\overrightarrow{e}}_y^a{\overrightarrow{e}}_x^b={\overrightarrow{e}}_y^b{\overrightarrow{e}}_x^a=\sin \left({\phi }_{xy}^b-{\phi }_{xy}^a\right).$$

(29)

In (29), it is taken into account that the angles are also measured from the unit vector, which is indicated by the first factor.

For orthogonal products of unit vectors, we take

$${\phi }_{xy}^b-{\phi }_{xy}^a=\left(\pm 1\right)\frac{\pi }{2}=\frac{\pi }{2}{\overrightarrow{e}}_z.$$

(30)

Then, from Equations (28) and (29) it follows

$${\overrightarrow{e}}_x^a{\overrightarrow{e}}_x^b={\overrightarrow{e}}_x^b{\overrightarrow{e}}_x^a={\overrightarrow{e}}_y^b{\overrightarrow{e}}_y^a={\overrightarrow{e}}_y^a{\overrightarrow{e}}_y^b=0,$$

(31)

$${\overrightarrow{e}}_x^a{\overrightarrow{e}}_y^b=-{\overrightarrow{e}}_x^b{\overrightarrow{e}}_y^a=-{\overrightarrow{e}}_y^a{\overrightarrow{e}}_x^b={\overrightarrow{e}}_y^b{\overrightarrow{e}}_x^a={\overrightarrow{e}}_z.$$

(32)

In Equation (30), as in traditional vector algebra, a new unit vector ${\overrightarrow{e}}_z$ is introduced for the linear display of angular directions.

As shown above, the application of Equations (31) and (32) when multiplying noncollinear factors makes it possible to ensure the commutativity of the product of vectors and the commutativity of the algebra of hypercomplex numbers.

5. Conclusions

Until now, in all textbooks and scientific works on algebra, it has been argued that, unlike real and complex numbers, the construction of a commutative division algebra of quaternions over the field of real numbers is impossible in principle. Almost no one questioned the existing theoretical statement that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property.

This article demonstrates for the first time:

(1) The possibility of constructing a normed commutative quaternion algebra with division over the field of real numbers;

(2) The possibility of constructing a normed commutative octonion algebra with division over the field of real numbers;

(3) The possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers;

(4) The possibility of constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers;

(5) A method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N;

(6) The possibility of a significant expansion of the list of algebras that satisfy the sum of squares identity for normed division algebras over the field of real numbers;

(7) The product of vectors has the property of commutativity; when multiplying, it is necessary to use specific forms of representation of factors.

The normed commutative algebras of N-dimensional hypercomplex numbers can be widely used to solve many topical scientific problems in various areas of fundamental and applied sciences. For example, in the field of theoretical physics, one can successfully model force fields; in cryptography, one can propose a whole range of new cryptographic programs using hypercomplex number algebras with different dimension values. There are many other possibilities for applying the results of this work.