First Exploration of a Novel Generalization of Lie and Malcev Algebras Leading to the Emergence of Complex Structures

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First Exploration of a Novel Generalization of Lie and Malcev Algebras Leading to the Emergence of Complex Structures

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Alejandro Jesús Bermejo Valdés^*

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Submitted:

15 June 2024

Posted:

17 June 2024

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Abstract

We introduce a novel algebraic structure, termed algebra B, which generalizes Lie algebras through the definition of a non-associative and unital algebra. We demonstrate that algebra B retains essential properties such as bilinearity and antisymmetry, and satisfies both the Jacobi and Malcev identities; this occurs when the product operations of Lie and Malcev algebras are derived from the product operation defined in our algebra. By examining the connections between algebra B, Malcev algebras, and Lie algebras, we establish that algebra B effectively generalizes these structures. Furthermore, we show that algebra B can generate complex entities with a novel structure distinct from those currently known. Our findings lay the groundwork for future investigations into the practical applications and further theoretical development of this new algebraic framework.

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction

Lie algebras are fundamental in both mathematics and theoretical physics due to their capacity to describe and analyze symmetries and continuous transformations. They find key applications in group theory, differential geometry, and representation theory, providing essential tools for studying algebraic and geometric structures. In theoretical physics, Lie algebras are crucial in quantum mechanics, quantum field theory, and general relativity, where they help understand the fundamental symmetries of physical laws and the behavior of particles and fields. This versatility and explanatory power make Lie algebras a central topic in advanced research [1,2].

The pursuit of generalizing mathematical definitions and exploring increasingly fundamental entities is a cornerstone of mathematical research. Lie algebras serve as a bridge between quantum mechanics and general relativity [2,3], motivating the search for a “unifying algebra” capable of integrating these two pivotal theories.

In this study, we present a comprehensive analysis of a novel algebraic structure, denoted as algebra B, designed to generalize Lie algebras through the definition of a non-associative and unital algebra. We demonstrate that the algebra B preserves essential algebraic properties such as bilinearity and antisymmetry, and satisfies both the Jacobi and Malcev identities when the products defining these algebras are derived from the product of the algebra B. By examining the interrelations between algebra B, Malcev algebras, and Lie algebras, we establish that algebra B functions as a valid and broad generalization of these structures. Moreover, we reveal that algebra B possesses the remarkable ability to generate imaginary units and complex entities with unique structures, distinct from those currently known. These findings open new avenues for theoretical investigation and practical application, providing a framework for future research in algebraic symmetries and transformations.

1.1. Lie Algebra

A Lie algebra is an algebraic structure consisting of a vector space g over a field K, together with a binary operation called the Lie bracket, denoted by

[,] : g \times g \to g

, which satisfies the following properties [3,4,5]:

Bilinearity: $\forall λ, γ \in K; \forall a, b, c \in g : [λ a + γ b, c] = λ [a, c] + γ [b, c]$ and $[c, λ a + γ b] = λ [c, a] + γ [c, b]$ .
Antisymmetry: $\forall a, b \in g : [a, b] = - [b, a]$ . This property implies that $\forall a \in g : [a, a] = 0$ .
Jacobi Identity: $[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0, \forall a, b, c \in g$ .

This structure is not necessarily associative, as the antisymmetry property and the Jacobi identity do not imply the associativity of the Lie bracket operation [3,4].

1.2. Malcev Algebras

Malcev algebras, which are a direct generalization of Lie algebras, form an important algebraic structure. While every Lie algebra is inherently a Malcev algebra, the converse is not true; not all Malcev algebras are Lie algebras. The defining feature of Malcev algebras is their satisfaction of a specific identity analogous to the Jacobi identity, known as the Malcev identity [6,7].

A Malcev algebra is defined as an algebraic structure comprising a vector space M over a field K, equipped with a binary operation

[,] : M \times M \to M

. This operation adheres to the following properties [6,7]:

Antisymmetry: $\forall a, b \in M : [a, b] = - [b, a]$ .
Malcev Identity: $\forall a, b, c \in M : [[a, b], [a, c]] = [[[a, b], c], a] + [[[b, c], a], a] + [[[c, a], a], b]$ .

The Malcev identity serves as a generalization of the Jacobi identity found in Lie algebras. Specifically, within the context of a Lie algebra, the Malcev identity simplifies to the Jacobi identity, highlighting the foundational relationship between these two algebraic structures [6,8].

Therefore, an algebra that aims to generalize both Lie and Malcev algebras must, by definition, satisfy the bilinearity of the product, antisymmetry, the Jacobi identity, and the Malcev identity.

2. Construction of Algebra $(V, ⊙)$

2.1. Algebraic Structure of the Vector Space V

Let the set V be a mathematical entity that satisfies the Zermelo-Fraenkel axioms and the Axiom of Choice [9,10].

Let the algebraic structure of the vector space be defined as

V \equiv (V^{3}, +, \cdot)

over the field F, where + is the addition operation in

V^{3}

and the operation · denotes scalar multiplication of the elements of

V^{3}

with the elements of the field F. The elements of V, which we will call vectors, are defined as ordered 3-tuples of the form

(v_{1}, v_{2}, v_{3})

such that

v_{i} \in F

. Thus, we have defined a vector space over a field F whose vectors contain entries in F [11,12]. The addition operation + is defined as:

\begin{matrix} + : V \times V \to V, \\ ((a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})) \mapsto (c_{1}, c_{2}, c_{3}), \\ (a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3}) \equiv (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}) . \end{matrix}

This operation, defined through elements of F, is associative and commutative. The identity element of the addition is

(0, 0, 0) \equiv {id}_{+}

, and the inverse elements correspond to the additive inverses in F[11]:

(a_{1}, a_{2}, a_{3}) + (- a_{1}, - a_{2}, - a_{3}) = {id}_{+}

. Thus, the substructure

(V^{3}, +)

is an abelian group [11,13].

The scalar multiplication operation · is defined as:

\begin{matrix} \cdot : F \times V \to V, \\ (λ, (a_{1}, a_{2}, a_{3})) \mapsto λ \cdot (a_{1}, a_{2}, a_{3}), \\ λ \cdot (a_{1}, a_{2}, a_{3}) \equiv (λ \cdot a_{1}, λ \cdot a_{2}, λ \cdot a_{3}), λ \in F . \end{matrix}

This operation is associative, commutative, distributive with respect to vector addition, and distributive with respect to addition in F[11]. The identity element corresponds to the identity in F:

{id}_{\cdot} \cdot (a_{1}, a_{2}, a_{3}) = (a_{1}, a_{2}, a_{3})

. Thus, the substructure

(V^{3}, \cdot)

is a commutative monoid [11,13].

Therefore, the complete structure

V \equiv (V^{3}, +, \cdot)

is a commutative ring [11,13].

2.2. Definition of the Algebra $(V, ⊙)$

We equip the structure V with a bilinear product operation ⊙, thus generating an F-algebra B that we denote as

B \equiv (V, ⊙)

Let the arbitrary 3-tuples

(a_{1}, a_{2}, a_{3})

and

(b_{1}, b_{2}, b_{3})

, we define the product operation: The operation ⊙ is defined as:

\begin{matrix} ⊙ : V \times V \to V, \\ ((a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})) \mapsto (c_{1}, c_{2}, c_{3}), \\ (a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) \equiv \\ \equiv (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}, a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}, a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) . \end{matrix}

In algebra B, the equality between elements is defined as follows:

\forall a, b \in F : (a_{1}, a_{2}, a_{3}) = (b_{1}, b_{2}, b_{3}) \Leftrightarrow a_{1} = b_{1} \land a_{2} = b_{2} \land a_{3} = b_{3} .

2.2.1. Bilinearity of the Product

Our definition of the product satisfies (see Appendix A) the following three properties of bilinearity:

Left distributivity with respect to addition +: $(a_{1}, a_{2}, a_{3}) ⊙ ((b_{1}, b_{2}, b_{3}) + (c_{1}, c_{2}, c_{3})) = (a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) + (a_{1}, a_{2}, a_{3}) ⊙ (c_{1}, c_{2}, c_{3})$ .
Right distributivity with respect to addition +: $((a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3})) ⊙ (c_{1}, c_{2}, c_{3}) = (a_{1}, a_{2}, a_{3}) ⊙ (c_{1}, c_{2}, c_{3}) + (b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3})$ .
Multiplication · of field elements with respect to ⊙: $\forall λ \in F : (λ \cdot (a_{1}, a_{2}, a_{3})) ⊙ (b_{1}, b_{2}, b_{3}) = (a_{1}, a_{2}, a_{3}) ⊙ (λ \cdot (b_{1}, b_{2}, b_{3})) = λ \cdot ((a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}))$ .

This last property arises directly from the commutativity of the entries of the vectors, which are elements of F, with respect to the elements of the field F of the algebra.

2.2.2. Basic Properties of Algebra B

2.2.2.1. Non-Associativity and Non-Alternativity

Our algebra exhibits non-associativity (see Appendix B), hence:

((a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3})) ⊙ (c_{1}, c_{2}, c_{3}) \neq (a_{1}, a_{2}, a_{3}) ⊙ ((b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3})) .

We define the Associator operator as

ASSOC : = (X ⊙ Y) ⊙ Z - X ⊙ (Y ⊙ Z)

, such that

X, Y, Z \in V

. According to this definition:

\begin{matrix} ASSOC & = ((a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3})) ⊙ (c_{1}, c_{2}, c_{3}) \\ - (a_{1}, a_{2}, a_{3}) ⊙ ((b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3})) \\ = (d, f, g), \end{matrix}

where:

d = - (a_{3} \cdot b_{2} \cdot c_{2}) - (a_{3} \cdot b_{2} \cdot c_{3}) + (a_{2} \cdot b_{3} \cdot c_{2}) + (a_{3} \cdot b_{3} \cdot c_{2})

f = (- a_{3} \cdot b_{3} \cdot c_{2}) + (a_{3} \cdot b_{2} \cdot c_{2}) + (a_{2} \cdot b_{3} \cdot c_{3}) - (a_{3} \cdot b_{3} \cdot c_{2})

, and

g = (- a_{2} \cdot b_{2} \cdot c_{3}) + (a_{3} \cdot b_{2} \cdot c_{2}) - (a_{3} \cdot b_{3} \cdot c_{2}) + (a_{3} \cdot b_{2} \cdot c_{2})

. This demonstrates that our algebra is a non-associative F-algebra.

A non-associative algebra can be an alternative algebra if it satisfies

X (X Y) = (X X) Y

and

(Y X) X = Y (X X)

for all

X, Y

in the algebra [14,15]. We examine this property because an alternative algebra may be made into a Malcev algebra by defining the Malcev product as

[X, Y] \equiv X Y - Y X

[6,8,14,15].

We explored both properties:

X (X Y) = (X X) Y

and

(Y X) X = Y (X X)

We verified that the property

X (X Y) = (X X) Y

does not hold in our algebra (see Appendix C), and for this property, we defined an Alternativator operator,

{ALTER}_{1}

, such that:

{ALTER}_{1} \equiv X ⊙ (X ⊙ Y) - (X ⊙ X) ⊙ Y .

Our Alternativator

{ALTER}_{1}

resulted in the following:

\begin{matrix} {ALTER}_{1} & = (- a_{3} \cdot a_{3} \cdot b_{2} + a_{3} \cdot a_{2} \cdot b_{3}, \\ - a_{2} \cdot a_{3} \cdot b_{3} + a_{3} \cdot a_{3} \cdot b_{2} + a_{3} \cdot a_{3} \cdot b_{2} - a_{3} \cdot a_{2} \cdot b_{2}, \\ a_{3} \cdot a_{3} \cdot b_{2} - a_{3} \cdot a_{2} \cdot b_{2} + a_{2} \cdot a_{2} \cdot b_{3} - a_{3} \cdot a_{2} \cdot b_{2}) . \end{matrix}

The property

(Y X) X = Y (X X)

also did not hold in our algebra (see Appendix C). For this case, we defined the Alternativator operator,

{ALTER}_{2}

, as follows:

{ALTER}_{2} \equiv (Y ⊙ X) ⊙ Y - Y ⊙ (X ⊙ X) =

(- b_{2} \cdot a_{3} \cdot a_{2} + b_{3} \cdot a_{2} \cdot a_{2}, - b_{2} \cdot a_{3} \cdot a_{3} + b_{3} \cdot a_{3} \cdot a_{2} + b_{3} \cdot a_{3} \cdot a_{2} - b_{3} \cdot a_{2} \cdot a_{2}, b_{3} \cdot a_{3} \cdot a_{2} - b_{3} \cdot a_{2} \cdot a_{2} + b_{2} \cdot a_{2} \cdot a_{3} - b_{3} \cdot a_{2} \cdot a_{2}) .

Therefore, the algebra B is non-associative and non-alternative.

2.2.2.2. Non-Commutativity and Antisymmetry

We define the Commutator (or Bracket) operator

[X, Y] \equiv X ⊙ Y - Y ⊙ X

, where

X, Y \in V

. According to this definition:

[(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})] =

(a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) - (b_{1}, b_{2}, b_{3}) ⊙ (a_{1}, a_{2}, a_{3}) = (0, a_{3} \cdot b_{2} - b_{3} \cdot a_{2}, a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) = β \cdot (0, 1, 1),

where

β \equiv a_{3} \cdot b_{2} - b_{3} \cdot a_{2}

, 1 represents the identity element for multiplication · in the field F, and 0 denotes the identity element for addition in F. This demonstrates that our algebra is non-commutative.

Note that the commutator results in the cancellation of the first component. Furthermore, it reveals the field element

β = a_{3} \cdot b_{2} - b_{3} \cdot a_{2}

. In this way, our commutator has the structure of a determinant:

- det (u, a, b) = - |\begin{matrix} 1 & 0 & 0 \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix}| = - |\begin{matrix} a_{2} & a_{3} \\ b_{2} & b_{3} \end{matrix}| = a_{3} \cdot b_{2} - a_{2} \cdot b_{3},

where

u \equiv (1, 0, 0)

a \equiv (a_{1}, a_{2}, a_{3})

and

b \equiv (b_{1}, b_{2}, b_{3})

Therefore, we can define our commutator generically as:

[(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})] = - det (u, a, b) \cdot (0, 1, 1),

If we evaluate the commutator of the same element, we have:

[(a_{1}, a_{2}, a_{3}), (a_{1}, a_{2}, a_{3})] = (a_{1}, a_{2}, a_{3}) ⊙ (a_{1}, a_{2}, a_{3}) - (a_{1}, a_{2}, a_{3}) ⊙ (a_{1}, a_{2}, a_{3}) = (0, 0, 0) .

On the other hand, let’s analyze the following commutators:

[(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})] = (0, a_{3} \cdot b_{2} - b_{3} \cdot a_{2}, a_{3} \cdot b_{2} - b_{3} \cdot a_{2}),

[(b_{1}, b_{2}, b_{3}), (a_{1}, a_{2}, a_{3})] = - (0, a_{3} \cdot b_{2} - a_{2} \cdot b_{3}, a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) = - [(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})] .

Thus:

[(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})] = - [(b_{1}, b_{2}, b_{3}), (a_{1}, a_{2}, a_{3})],

where this property manifests as the antisymmetry of the operation

[X, Y]

when defined via the product ⊙ in algebra B.

Note that algebra B itself is not antisymmetric, due to:

(a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) \neq - (b_{1}, b_{2}, b_{3}) ⊙ (a_{1}, a_{2}, a_{3}) .

2.2.2.3. Identity Element

The identity element of the product ⊙ in B is

(1, 0, 0)

. According to this, we have:

(a_{1}, a_{2}, a_{3}) ⊙ (1, 0, 0) = (1, 0, 0) ⊙ (a_{1}, a_{2}, a_{3}) = (a_{1}, a_{2}, a_{3}) .

Knowing that

u \equiv (1, 0, 0)

, we can redefine the commutator as:

[(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})] = - det ({id}_{⊙}, a, b) \cdot (0, 1, 1),

where we have redefined u as

{id}_{⊙} \equiv (1, 0, 0)

The existence of an identity element makes our algebra a unital algebra.

2.3. Relation of Algebra B with Malcev and Lie Algebras

Our algebra is non-commutative and non-associative. Due to their structural properties, Malcev and Lie algebras could be related to our algebra. To verify the similarities between algebra B and both Malcev and Lie algebras, we need to demonstrate that algebra B satisfies the defining properties of Malcev and Lie algebras.

2.3.1. Relation of Algebra B with Lie Algebra

To demonstrate that a Lie algebra derives from our algebra B, it is necessary to show that the Lie bracket defined from our product ⊙ satisfies the properties of bilinearity and the Jacobi identity, as antisymmetry has been previously demonstrated in section “2.2.2. Non-Commutativity and Antisymmetry”.

2.3.1.1. Bilinearity of the Commutator

We have established that the bilinearity of the Lie bracket, defined from the product ⊙, holds true (see Appendix D). This confirms that the following property is satisfied in our algebra:

\forall λ, γ \in F : [λ (a_{1}, a_{2}, a_{3}) + γ (b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] = λ [(a_{1}, a_{2}, a_{3}), (c_{1}, c_{2}, c_{3})] + γ [(b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] .

2.3.1.2. Jacobi Identity

The Jacobi identity [3,4] holds in algebra B when we define the Lie bracket through our product ⊙. Therefore, the Jacobi identity is a property that represents a particular case in algebra B (see Appendix E).

2.3.2. Construction of Malcev Algebra through Algebra B

The Malcev identity is satisfied if

\forall a, b, c \in M : [[a, b], [a, c]] = [[[a, b], c], a] + [[[b, c], a], a] + [[[c, a], a], b]

[6,7].

We verify that the algebra B satisfies the Malcev identity by defining the brackets in terms of the product ⊙ (see Appendix F). In fact, it satisfies the identity in a particular way; in our algebra, this identity has the following structure:

\forall a, b, c \in V : [[a, b], [a, c]] = [[[a, b], c], a] + [[[b, c], a], a] + [[[c, a], a], b] = (0, 0, 0) .

Thus, for algebra B, the following identities hold:

[[[a, b], c], a] = - [[[b, c], a], a] - [[[c, a], a], b],

[[[b, c], a], a] = - [[[a, b], c], a] - [[[c, a], a], b],

[[[c, a], a], b] = - [[[a, b], c], a] - [[[b, c], a], a] .

3. Generation of Complex Entities from B

We extracted other basic properties of the algebra B by evaluating the product of several vectors with particular structures. From this analysis, imaginary units emerged for our algebra. Thus, the algebra B is capable of generating complex entities.

3.1. Imaginary Unit i in B

Let the product

(0, a_{2}, 0) ⊙ (0, b_{2}, 0) = (- a_{2} \cdot b_{2}, 0, 0)

. In this way, for the case

a_{2} = b_{2} = 1

, we have:

(0, 1, 0) ⊙ (0, 1, 0) = (- 1, 0, 0) = - {id}_{⊙} .

Which, if we define

i \equiv (0, 1, 0)

and

i^{2} \equiv i ⊙ i

results in:

i^{2} = - {id}_{⊙} .

Note also that

(- 1, 0, 0) ⊙ (- 1, 0, 0) = (1, 0, 0) = {id}_{⊙}

3.2. Imaginary Unit j in B

Let the product

(0, 0, a_{3}) ⊙ (0, 0, b_{3}) = (- a_{3} \cdot b_{3}, 0, 0)

. In this way, if

a_{3} = b_{3} = 1

, we have:

(0, 0, 1) ⊙ (0, 0, 1) = - {id}_{⊙} .

Defining

j \equiv (0, 0, 1)

results in:

j^{2} = - {id}_{⊙} .

We then define the complex number

b = (b_{1}, b_{2}, b_{3})

such that:

b_{1} = Re (b),

b_{2} = {Im}_{1} (b),

b_{3} = {Im}_{2} (b),

where

Re (b)

is read as “real part of b”,

{Im}_{1} (b)

is read as “first imaginary part of b”, and

{Im}_{2} (b)

is read as “second imaginary part of b”.

Consistent with this, we may consider the elements

(b_{1}, b_{2}, b_{3})

of the algebra B as complex entities, such that the following expression can be defined:

b = (b_{1}, b_{2}, b_{3}) = b_{1} \cdot (1, 0, 0) + b_{2} \cdot (0, 1, 0) + b_{3} \cdot (0, 0, 1) = b_{1} + b_{2} \cdot i + b_{3} \cdot j, i^{2} = j^{2} = - {id}_{⊙} .

4. Other Properties

4.1. Nullification of the Product $a \cdot i ⊙ b \cdot j$

In our algebra, the product of the form

(0, a_{2}, 0) ⊙ (0, 0, b_{3})

results in a null vector,

0 \equiv (0, 0, 0)

. Therefore,

a \cdot i ⊙ b \cdot j = 0 .

We will call this property “orthomulearity”, simply because it resembles the orthogonality of vector spaces under the inner product. In our algebra, when two elements of the type

a \cdot i

and

b \cdot j

are multiplied with ⊙, the result nullifies regardless of the factors involved.

4.2. Product $a_{3} \cdot j ⊙ b_{2} \cdot i$

Let

(0, 0, a_{3}) ⊙ (0, b_{2}, 0)

, we have:

(0, 0, a_{3}) ⊙ (0, b_{2}, 0) = (0, a_{3} \cdot b_{2}, a_{3} \cdot b_{2}) = a_{3} \cdot b_{2} \cdot (0, 1, 1),

and, for

a_{3} = b_{2} = 1

(0, 0, 1) ⊙ (0, 1, 0) = (0, 1, 1) .

Writing

(0, 0, a_{3}) ⊙ (0, b_{2}, 0)

in the form

a_{3} \cdot j ⊙ b_{2} \cdot i

, we have:

a_{3} \cdot j ⊙ b_{2} \cdot i = (a_{3} \cdot b_{2}) \cdot j ⊙ i = (a_{3} \cdot b_{2}) \cdot (0, 1, 1) = (a_{3} \cdot b_{2}) \cdot ((0, 0, 1) ⊙ (0, 1, 0)) .

5. Conclusions

We have introduced a novel algebraic structure termed algebra B, which generalizes Lie algebras through the definition of a non-associative and unital algebra. Our results demonstrate that algebra B retains essential properties such as bilinearity and antisymmetry, and satisfies both the Jacobi and Malcev identities when we define the product of their corresponding algebras through the product of our algebra.

We have shown that algebra B can be considered a valid generalization of Malcev and Lie algebras, highlighting its potential to integrate fundamental algebraic structures. Additionally, the ability of algebra B to generate imaginary units and complex entities opens new avenues for research in algebraic symmetries and transformations.

Future research should focus on exploring its practical applications and extending the theoretical framework established here.

Our study significantly contributes to the field of algebra theory, offering a new perspective and potential tools for the investigation of algebraic structures and their applications across various scientific disciplines. This research sets the stage for future explorations in the quest for a unifying algebra that can integrate key concepts in mathematics and theoretical physics.

Appendix A. Proof of the Bilinearity of the Product ⊙

Left distributivity with respect to addition +

(a_{1}, a_{2}, a_{3}) ⊙ ((b_{1}, b_{2}, b_{3}) + (c_{1}, c_{2}, c_{3})) = (a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) + (a_{1}, a_{2}, a_{3}) ⊙ (c_{1}, c_{2}, c_{3}) .

Left-hand side: Expanding +:

\begin{matrix} (a_{1}, a_{2}, a_{3}) ⊙ ((b_{1}, b_{2}, b_{3}) + (c_{1}, c_{2}, c_{3})) & = (a_{1}, a_{2}, a_{3}) ⊙ (b_{1} + c_{1}, b_{2} + c_{2}, b_{3} + c_{3}) \\ = (a_{1} \cdot (b_{1} + c_{1}) - a_{2} \cdot (b_{2} + c_{2}) - a_{3} \cdot (b_{3} + c_{3}), \\ a_{1} \cdot (b_{2} + c_{2}) + a_{2} \cdot (b_{1} + c_{1}) + a_{3} \cdot (b_{2} + c_{2}), \\ a_{1} \cdot (b_{3} + c_{3}) + a_{3} \cdot (b_{2} + c_{2}) + a_{3} \cdot (b_{1} + c_{1})) \\ = (a_{1} \cdot b_{1} + a_{1} \cdot c_{1} - a_{2} \cdot b_{2} - a_{2} \cdot c_{2} - a_{3} \cdot b_{3} - a_{3} \cdot c_{3}, \\ a_{1} \cdot b_{2} + a_{1} \cdot c_{2} + a_{2} \cdot b_{1} + a_{2} \cdot c_{1} + a_{3} \cdot b_{2} + a_{3} \cdot c_{2}, \\ a_{1} \cdot b_{3} + a_{1} \cdot c_{3} + a_{3} \cdot b_{2} + a_{3} \cdot c_{2} + a_{3} \cdot b_{1} + a_{3} \cdot c_{1}) . \end{matrix}

Right-hand side:

\begin{matrix} (a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3}) + (a_{1}, a_{2}, a_{3}) ⊙ (c_{1}, c_{2}, c_{3}) & = (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3} + a_{1} \cdot c_{1} - a_{2} \cdot c_{2} - a_{3} \cdot c_{3}, \\ a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2} + a_{1} \cdot c_{2} + a_{2} \cdot c_{1} + a_{3} \cdot c_{2}, \\ a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1} + a_{1} \cdot c_{3} + a_{3} \cdot c_{2} + a_{3} \cdot c_{1}) . \end{matrix}

Left-hand side = Right-hand side.

Right distributivity with respect to addition +

((a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3})) ⊙ (c_{1}, c_{2}, c_{3}) = (a_{1}, a_{2}, a_{3}) ⊙ (c_{1}, c_{2}, c_{3}) + (b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3}) .

Left-hand side:

\begin{matrix} ((a_{1}, a_{2}, a_{3}) + (b_{1}, b_{2}, b_{3})) ⊙ (c_{1}, c_{2}, c_{3}) & = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3}) ⊙ (c_{1}, c_{2}, c_{3}) \\ = ((a_{1} + b_{1}) \cdot c_{1} - (a_{2} + b_{2}) \cdot c_{2} - (a_{3} + b_{3}) \cdot c_{3}, \\ (a_{1} + b_{1}) \cdot c_{2} + (a_{2} + b_{2}) \cdot c_{1} + (a_{3} + b_{3}) \cdot c_{2}, \\ (a_{1} + b_{1}) \cdot c_{3} + (a_{3} + b_{3}) \cdot c_{2} + (a_{3} + b_{3}) \cdot c_{1}) \\ = (a_{1} \cdot c_{1} + b_{1} \cdot c_{1} - a_{2} \cdot c_{2} - b_{2} \cdot c_{2} - a_{3} \cdot c_{3} - b_{3} \cdot c_{3}, \\ a_{1} \cdot c_{2} + b_{1} \cdot c_{2} + a_{2} \cdot c_{1} + b_{2} \cdot c_{1} + a_{3} \cdot c_{2} + b_{3} \cdot c_{2}, \\ a_{1} \cdot c_{3} + b_{1} \cdot c_{3} + a_{3} \cdot c_{2} + b_{3} \cdot c_{2} + a_{3} \cdot c_{1} + b_{3} \cdot c_{1}) . \end{matrix}

Right-hand side:

\begin{matrix} (a_{1}, a_{2}, a_{3}) ⊙ (c_{1}, c_{2}, c_{3}) + (b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3}) & = (a_{1} \cdot c_{1} - a_{2} \cdot c_{2} - a_{3} \cdot c_{3}, \\ a_{1} \cdot c_{2} + a_{2} \cdot c_{1} + a_{3} \cdot c_{2}, \\ a_{1} \cdot c_{3} + a_{3} \cdot c_{2} + a_{3} \cdot c_{1}) + (b_{1} \cdot c_{1} - b_{2} \cdot c_{2} - b_{3} \cdot c_{3}, \\ b_{1} \cdot c_{2} + b_{2} \cdot c_{1} + b_{3} \cdot c_{2}, \\ b_{1} \cdot c_{3} + b_{3} \cdot c_{2} + b_{3} \cdot c_{1}) \\ = (a_{1} \cdot c_{1} - a_{2} \cdot c_{2} - a_{3} \cdot c_{3} + b_{1} \cdot c_{1} - b_{2} \cdot c_{2} - b_{3} \cdot c_{3}, \\ a_{1} \cdot c_{2} + a_{2} \cdot c_{1} + a_{3} \cdot c_{2} + b_{1} \cdot c_{2} + b_{2} \cdot c_{1} + b_{3} \cdot c_{2}, \\ a_{1} \cdot c_{3} + a_{3} \cdot c_{2} + a_{3} \cdot c_{1} + b_{1} \cdot c_{3} + b_{3} \cdot c_{2} + b_{3} \cdot c_{1}) . \end{matrix}

Left-hand side = Right-hand side.

Multiplication · of field elements with respect to ⊙

(λ \cdot (a_{1}, a_{2}, a_{3})) ⊙ (b_{1}, b_{2}, b_{3}) = (a_{1}, a_{2}, a_{3}) ⊙ (λ \cdot (b_{1}, b_{2}, b_{3})) = λ \cdot ((a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3})), λ \in F .

This last property arises directly from the commutativity that exists in the entries of the vectors, which are elements of F, with respect to the elements of the field F of the algebra.

Appendix B. Proof of the Non-Associativity of the Product ⊙

We initially evaluate:

\begin{matrix} ((a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3})) ⊙ (c_{1}, c_{2}, c_{3}) & = ((a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}, \\ a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}, \\ a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) ⊙ (c_{1}, c_{2}, c_{3})) \\ = ((a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}) \cdot c_{1} \\ - (a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}) \cdot c_{2} \\ - (a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) \cdot c_{3}, \\ (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}) \cdot c_{2} \\ + (a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}) \cdot c_{1} \\ + (a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) \cdot c_{2}, \\ (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}) \cdot c_{3} \\ + (a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) \cdot c_{2} \\ + (a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) \cdot c_{1}) . \end{matrix}

Now we evaluate:

\begin{matrix} (a_{1}, a_{2}, a_{3}) ⊙ ((b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3})) & = (a_{1}, a_{2}, a_{3}) ⊙ (b_{1} \cdot c_{1} - b_{2} \cdot c_{2} - b_{3} \cdot c_{3}, \\ b_{1} \cdot c_{2} + b_{2} \cdot c_{1} + b_{3} \cdot c_{2}, \\ b_{1} \cdot c_{3} + b_{3} \cdot c_{2} + b_{3} \cdot c_{1}) \\ = (a_{1} \cdot (b_{1} \cdot c_{1} - b_{2} \cdot c_{2} - b_{3} \cdot c_{3}) \\ - a_{2} \cdot (b_{1} \cdot c_{2} + b_{2} \cdot c_{1} + b_{3} \cdot c_{2}) \\ - a_{3} \cdot (b_{1} \cdot c_{3} + b_{3} \cdot c_{2} + b_{3} \cdot c_{1}), \\ a_{1} \cdot (b_{1} \cdot c_{2} + b_{2} \cdot c_{1} + b_{3} \cdot c_{2}) \\ + a_{2} \cdot (b_{1} \cdot c_{1} - b_{2} \cdot c_{2} - b_{3} \cdot c_{3}) \\ + a_{3} \cdot (b_{1} \cdot c_{2} + b_{2} \cdot c_{1} + b_{3} \cdot c_{2}), \\ a_{1} \cdot (b_{1} \cdot c_{3} + b_{3} \cdot c_{2} + b_{3} \cdot c_{1}) \\ + a_{3} \cdot (b_{1} \cdot c_{2} + b_{2} \cdot c_{1} + b_{3} \cdot c_{2}) \\ + a_{3} \cdot (b_{1} \cdot c_{1} - b_{2} \cdot c_{2} - b_{3} \cdot c_{3})) . \end{matrix}

Comparing component by component:

First component:

\begin{matrix} (a_{1} \cdot b_{1} \cdot c_{1} - a_{2} \cdot b_{2} \cdot c_{1} - a_{3} \cdot b_{3} \cdot c_{1}) \\ - (a_{1} \cdot b_{2} \cdot c_{2} + a_{2} \cdot b_{1} \cdot c_{2} + a_{3} \cdot b_{2} \cdot c_{2}) \\ - (a_{1} \cdot b_{3} \cdot c_{3} + a_{3} \cdot b_{2} \cdot c_{3} + a_{3} \cdot b_{1} \cdot c_{3}) \\ \neq (a_{1} \cdot b_{1} \cdot c_{1} - a_{1} \cdot b_{2} \cdot c_{2} - a_{1} \cdot b_{3} \cdot c_{3}) \\ - (a_{2} \cdot b_{1} \cdot c_{2} + a_{2} \cdot b_{2} \cdot c_{1} + a_{2} \cdot b_{3} \cdot c_{2}) \\ - (a_{3} \cdot b_{1} \cdot c_{3} + a_{3} \cdot b_{3} \cdot c_{2} + a_{3} \cdot b_{3} \cdot c_{1}) . \end{matrix}

\begin{matrix} - (a_{3} \cdot b_{2} \cdot c_{2}) - (a_{3} \cdot b_{2} \cdot c_{3}) \\ \neq - (a_{2} \cdot b_{3} \cdot c_{2}) - (a_{3} \cdot b_{3} \cdot c_{2}) . \end{matrix}

Second component:

\begin{matrix} (a_{1} \cdot b_{1} \cdot c_{2} - a_{2} \cdot b_{2} \cdot c_{2} - a_{3} \cdot b_{3} \cdot c_{2}) \\ + (a_{1} \cdot b_{2} \cdot c_{1} + a_{2} \cdot b_{1} \cdot c_{1} + a_{3} \cdot b_{2} \cdot c_{1}) \\ + (a_{1} \cdot b_{3} \cdot c_{2} + a_{3} \cdot b_{2} \cdot c_{2} + a_{3} \cdot b_{1} \cdot c_{2}) \\ \neq (a_{1} \cdot b_{1} \cdot c_{2} + a_{1} \cdot b_{2} \cdot c_{1} + a_{1} \cdot b_{3} \cdot c_{2}) \\ + (a_{2} \cdot b_{1} \cdot c_{1} - a_{2} \cdot b_{2} \cdot c_{2} - a_{2} \cdot b_{3} \cdot c_{3}) \\ + (a_{3} \cdot b_{1} \cdot c_{2} + a_{3} \cdot b_{2} \cdot c_{1} + a_{3} \cdot b_{3} \cdot c_{2}) . \end{matrix}

\begin{matrix} (- a_{3} \cdot b_{3} \cdot c_{2}) + (a_{3} \cdot b_{2} \cdot c_{2}) \\ \neq (- a_{2} \cdot b_{3} \cdot c_{3}) + (a_{3} \cdot b_{3} \cdot c_{2}) . \end{matrix}

Third component:

\begin{matrix} (a_{1} \cdot b_{1} \cdot c_{3} - a_{2} \cdot b_{2} \cdot c_{3} - a_{3} \cdot b_{3} \cdot c_{3}) \\ + (a_{1} \cdot b_{3} \cdot c_{2} + a_{3} \cdot b_{2} \cdot c_{2} + a_{3} \cdot b_{1} \cdot c_{2}) \\ + (a_{1} \cdot b_{3} \cdot c_{1} + a_{3} \cdot b_{2} \cdot c_{1} + a_{3} \cdot b_{1} \cdot c_{1}) \\ \neq (a_{1} \cdot b_{1} \cdot c_{3} + a_{1} \cdot b_{3} \cdot c_{2} + a_{1} \cdot b_{3} \cdot c_{1}) \\ + (a_{3} \cdot b_{1} \cdot c_{2} + a_{3} \cdot b_{2} \cdot c_{1} + a_{3} \cdot b_{3} \cdot c_{2}) \\ + (a_{3} \cdot b_{1} \cdot c_{1} - a_{3} \cdot b_{2} \cdot c_{2} - a_{3} \cdot b_{3} \cdot c_{3}) . \end{matrix}

\begin{matrix} (- a_{2} \cdot b_{2} \cdot c_{3}) + (a_{3} \cdot b_{2} \cdot c_{2}) \\ \neq (a_{3} \cdot b_{3} \cdot c_{2}) + (- a_{3} \cdot b_{2} \cdot c_{2}) . \end{matrix}

In this way:

\begin{matrix} ((a_{1}, a_{2}, a_{3}) ⊙ (b_{1}, b_{2}, b_{3})) ⊙ (c_{1}, c_{2}, c_{3}) \\ - (a_{1}, a_{2}, a_{3}) ⊙ ((b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3})) \\ = (- (a_{3} \cdot b_{2} \cdot c_{2}) - (a_{3} \cdot b_{2} \cdot c_{3}) + (a_{2} \cdot b_{3} \cdot c_{2}) + (a_{3} \cdot b_{3} \cdot c_{2}), \\ (- a_{3} \cdot b_{3} \cdot c_{2}) + (a_{3} \cdot b_{2} \cdot c_{2}) + (a_{2} \cdot b_{3} \cdot c_{3}) - (a_{3} \cdot b_{3} \cdot c_{2}), \\ (- a_{2} \cdot b_{2} \cdot c_{3}) + (a_{3} \cdot b_{2} \cdot c_{2}) - (a_{3} \cdot b_{3} \cdot c_{2}) + (a_{3} \cdot b_{2} \cdot c_{2})) . \end{matrix}

Appendix C. Proof of the Non-Alternativity of the Product ⊙

A non-associative algebra is alternative if:

X (X Y) = (X X) Y

and

(Y X) X = Y (X X)

for all

X, Y

in the algebra. Defining

X \equiv (a_{1}, a_{2}, a_{3})

and

Y \equiv (b_{1}, b_{2}, b_{3})

, we have:

X (X Y)

(a_{1}, a_{2}, a_{3}) ⊙ (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}, a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}, a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) =

\begin{matrix} (a_{1} \cdot (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}) \\ - a_{2} \cdot (a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}) \\ - a_{3} \cdot (a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}), \\ a_{1} \cdot (a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}) \\ + a_{2} \cdot (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3}) \\ + a_{3} \cdot (a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}), \\ a_{1} \cdot (a_{1} \cdot b_{3} + a_{3} \cdot b_{2} + a_{3} \cdot b_{1}) \\ + a_{3} \cdot (a_{1} \cdot b_{2} + a_{2} \cdot b_{1} + a_{3} \cdot b_{2}) \\ + a_{3} \cdot (a_{1} \cdot b_{1} - a_{2} \cdot b_{2} - a_{3} \cdot b_{3})) . \end{matrix}

(X X) Y

(a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3}, a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}, a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}) ⊙ (b_{1}, b_{2}, b_{3}) =

\begin{matrix} ((a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3}) \cdot b_{1} \\ - (a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}) \cdot b_{2} \\ - (a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}) \cdot b_{3}, \\ (a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3}) \cdot b_{2} \\ + (a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}) \cdot b_{1} \\ + (a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}) \cdot b_{2}, \\ (a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3}) \cdot b_{3} \\ + (a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}) \cdot b_{2} \\ + (a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}) \cdot b_{1}) . \end{matrix}

X (X Y) - (X X) Y

X (X Y) - (X X) Y =

(- a_{3} \cdot a_{3} \cdot b_{2} + a_{3} \cdot a_{2} \cdot b_{3}, - a_{2} \cdot a_{3} \cdot b_{3} + a_{3} \cdot a_{3} \cdot b_{2} + a_{3} \cdot a_{3} \cdot b_{2} - a_{3} \cdot a_{2} \cdot b_{2}, a_{3} \cdot a_{3} \cdot b_{2} - a_{3} \cdot a_{2} \cdot b_{2} + a_{2} \cdot a_{2} \cdot b_{3} - a_{3} \cdot a_{2} \cdot b_{2}) .

Y (X X)

(b_{1}, b_{2}, b_{3}) ⊙ (a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3}, a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}, a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}) =

\begin{matrix} (b_{1} \cdot (a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3}) \\ - b_{2} \cdot (a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}) \\ - b_{3} \cdot (a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}), \\ b_{1} \cdot (a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}) \\ + b_{2} \cdot (a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3}) \\ + b_{3} \cdot (a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}), \\ b_{1} \cdot (a_{1} \cdot a_{3} + a_{3} \cdot a_{2} + a_{3} \cdot a_{1}) \\ + b_{2} \cdot (a_{1} \cdot a_{2} + a_{2} \cdot a_{1} + a_{3} \cdot a_{2}) \\ + b_{3} \cdot (a_{1} \cdot a_{1} - a_{2} \cdot a_{2} - a_{3} \cdot a_{3})) . \end{matrix}

(Y X) X

(b_{1} \cdot a_{1} - b_{2} \cdot a_{2} - b_{3} \cdot a_{3}, b_{1} \cdot a_{2} + b_{2} \cdot a_{1} + b_{3} \cdot a_{2}, b_{1} \cdot a_{3} + b_{3} \cdot a_{2} + b_{3} \cdot a_{1}) ⊙ (a_{1}, a_{2}, a_{3}) =

\begin{matrix} ((b_{1} \cdot a_{1} - b_{2} \cdot a_{2} - b_{3} \cdot a_{3}) \cdot a_{1} \\ - (b_{1} \cdot a_{2} + b_{2} \cdot a_{1} + b_{3} \cdot a_{2}) \cdot a_{2} \\ - (b_{1} \cdot a_{3} + b_{3} \cdot a_{2} + b_{3} \cdot a_{1}) \cdot a_{3}, \\ (b_{1} \cdot a_{1} - b_{2} \cdot a_{2} - b_{3} \cdot a_{3}) \cdot a_{2} \\ + (b_{1} \cdot a_{2} + b_{2} \cdot a_{1} + b_{3} \cdot a_{2}) \cdot a_{1} \\ + (b_{1} \cdot a_{3} + b_{3} \cdot a_{2} + b_{3} \cdot a_{1}) \cdot a_{2}, \\ (b_{1} \cdot a_{1} - b_{2} \cdot a_{2} - b_{3} \cdot a_{3}) \cdot a_{3} \\ + (b_{1} \cdot a_{3} + b_{3} \cdot a_{2} + b_{3} \cdot a_{1}) \cdot a_{2} \\ + (b_{1} \cdot a_{3} + b_{3} \cdot a_{2} + b_{3} \cdot a_{1}) \cdot a_{1}) . \end{matrix}

(Y X) X - Y (X X)

(Y X) X - Y (X X) =

(- b_{2} \cdot a_{3} \cdot a_{2} + b_{3} \cdot a_{2} \cdot a_{2}, - b_{2} \cdot a_{3} \cdot a_{3} + b_{3} \cdot a_{3} \cdot a_{2} + b_{3} \cdot a_{3} \cdot a_{2} - b_{3} \cdot a_{2} \cdot a_{2}, b_{3} \cdot a_{3} \cdot a_{2} - b_{3} \cdot a_{2} \cdot a_{2} + b_{2} \cdot a_{2} \cdot a_{3} - b_{3} \cdot a_{2} \cdot a_{2}) .

Appendix D. Proof of the bilinearity of the Lie bracket defined from ⊙

Bilinearity holds if, for all

λ, γ \in F

[λ \cdot (a_{1}, a_{2}, a_{3}) + γ \cdot (b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] = λ \cdot [(a_{1}, a_{2}, a_{3}), (c_{1}, c_{2}, c_{3})] + γ \cdot [(b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] .

Left-hand side:

\begin{matrix} [λ \cdot (a_{1}, a_{2}, a_{3}) + γ \cdot (b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] = \\ λ \cdot (a_{1}, a_{2}, a_{3}) + γ \cdot (b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3}) - (c_{1}, c_{2}, c_{3}) ⊙ λ \cdot (a_{1}, a_{2}, a_{3}) + γ \cdot (b_{1}, b_{2}, b_{3}) . \end{matrix}

First term:

\begin{matrix} λ \cdot (a_{1}, a_{2}, a_{3}) + γ \cdot (b_{1}, b_{2}, b_{3}) ⊙ (c_{1}, c_{2}, c_{3}) \\ = (λ \cdot a_{1} + γ \cdot b_{1}, λ \cdot a_{2} + γ \cdot b_{2}, λ \cdot a_{3} + γ \cdot b_{3}) ⊙ (c_{1}, c_{2}, c_{3}) \\ = ((λ \cdot a_{1} + γ \cdot b_{1}) \cdot c_{1} - (λ \cdot a_{2} + γ \cdot b_{2}) \cdot c_{2} - (λ \cdot a_{3} + γ \cdot b_{3}) \cdot c_{3}, \\ (λ \cdot a_{1} + γ \cdot b_{1}) \cdot c_{2} + (λ \cdot a_{2} + γ \cdot b_{2}) \cdot c_{1} + (λ \cdot a_{3} + γ \cdot b_{3}) \cdot c_{2}, \\ (λ \cdot a_{1} + γ \cdot b_{1}) \cdot c_{3} + (λ \cdot a_{3} + γ \cdot b_{3}) \cdot c_{2} + (λ \cdot a_{3} + γ \cdot b_{3}) \cdot c_{1}) \\ = (λ \cdot a_{1} \cdot c_{1} + γ \cdot b_{1} \cdot c_{1} - λ \cdot a_{2} \cdot c_{2} - γ \cdot b_{2} \cdot c_{2} - λ \cdot a_{3} \cdot c_{3} - γ \cdot b_{3} \cdot c_{3}, \\ λ \cdot a_{1} \cdot c_{2} + γ \cdot b_{1} \cdot c_{2} + λ \cdot a_{2} \cdot c_{1} + γ \cdot b_{2} \cdot c_{1} + λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2}, \\ λ \cdot a_{1} \cdot c_{3} + γ \cdot b_{1} \cdot c_{3} + λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2} + λ \cdot a_{3} \cdot c_{1} + γ \cdot b_{3} \cdot c_{1}) . \end{matrix}

Second term:

\begin{matrix} (c_{1}, c_{2}, c_{3}) ⊙ λ \cdot (a_{1}, a_{2}, a_{3}) + γ \cdot (b_{1}, b_{2}, b_{3}) \\ = (c_{1}, c_{2}, c_{3}) ⊙ (λ \cdot a_{1} + γ \cdot b_{1}, λ \cdot a_{2}, γ \cdot b_{2}, λ \cdot a_{3}, γ \cdot b_{3}) \\ = (c_{1} \cdot (λ \cdot a_{1} + γ \cdot b_{1}) - c_{2} \cdot (λ \cdot a_{2} + γ \cdot b_{2}) - c_{3} \cdot (λ \cdot a_{3} + γ \cdot b_{3}), \\ c_{1} \cdot (λ \cdot a_{2} + γ \cdot b_{2}) + c_{2} \cdot (λ \cdot a_{1} + γ \cdot b_{1}) + c_{3} \cdot (λ \cdot a_{2} + γ \cdot b_{2}), \\ c_{1} \cdot (λ \cdot a_{3} + γ \cdot b_{3}) + c_{3} \cdot (λ \cdot a_{2} + γ \cdot b_{2}) + c_{3} \cdot (λ \cdot a_{1} + γ \cdot b_{1})) \\ = (c_{1} \cdot λ \cdot a_{1} + c_{1} \cdot γ \cdot b_{1} - c_{2} \cdot λ \cdot a_{2} - c_{2} \cdot γ \cdot b_{2} - c_{3} \cdot λ \cdot a_{3} - c_{3} \cdot γ \cdot b_{3}, \\ c_{1} \cdot λ \cdot a_{2} + c_{1} \cdot γ \cdot b_{2} + c_{2} \cdot λ \cdot a_{1} + c_{2} \cdot γ \cdot b_{1} + c_{3} \cdot λ \cdot a_{2} + c_{3} \cdot γ \cdot b_{2}, \\ c_{1} \cdot λ \cdot a_{3} + c_{1} \cdot γ \cdot b_{3} + c_{3} \cdot λ \cdot a_{2} + c_{3} \cdot γ \cdot b_{2} + c_{3} \cdot λ \cdot a_{1} + c_{3} \cdot γ \cdot b_{1}) . \end{matrix}

First term - second term:

\begin{matrix} (λ \cdot a_{1} \cdot c_{1} + γ \cdot b_{1} \cdot c_{1} - λ \cdot a_{2} \cdot c_{2} - γ \cdot b_{2} \cdot c_{2} - λ \cdot a_{3} \cdot c_{3} - γ \cdot b_{3} \cdot c_{3}, \\ λ \cdot a_{1} \cdot c_{2} + γ \cdot b_{1} \cdot c_{2} + λ \cdot a_{2} \cdot c_{1} + γ \cdot b_{2} \cdot c_{1} + λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2}, \\ λ \cdot a_{1} \cdot c_{3} + γ \cdot b_{1} \cdot c_{3} + λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2} + λ \cdot a_{3} \cdot c_{1} + γ \cdot b_{3} \cdot c_{1}) \\ - (c_{1} \cdot λ \cdot a_{1} + c_{1} \cdot γ \cdot b_{1} - c_{2} \cdot λ \cdot a_{2} - c_{2} \cdot γ \cdot b_{2} - c_{3} \cdot λ \cdot a_{3} - c_{3} \cdot γ \cdot b_{3}, \\ c_{1} \cdot λ \cdot a_{2} + c_{1} \cdot γ \cdot b_{2} + c_{2} \cdot λ \cdot a_{1} + c_{2} \cdot γ \cdot b_{1} + c_{3} \cdot λ \cdot a_{2} + c_{3} \cdot γ \cdot b_{2}, \\ c_{1} \cdot λ \cdot a_{3} + c_{1} \cdot γ \cdot b_{3} + c_{3} \cdot λ \cdot a_{2} + c_{3} \cdot γ \cdot b_{2} + c_{3} \cdot λ \cdot a_{1} + c_{3} \cdot γ \cdot b_{1}) \\ = (0, λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2} - c_{3} \cdot λ \cdot a_{2} - c_{3} \cdot γ \cdot b_{2}, \\ λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2} - c_{3} \cdot λ \cdot a_{2} - c_{3} \cdot γ \cdot b_{2}) . \end{matrix}

Right-hand side:

λ \cdot [(a_{1}, a_{2}, a_{3}), (c_{1}, c_{2}, c_{3})] + γ \cdot [(b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] .

First term:

\begin{matrix} λ \cdot [(a_{1}, a_{2}, a_{3}), (c_{1}, c_{2}, c_{3})] & = λ \cdot (a_{1} \cdot c_{1} - a_{2} \cdot c_{2} - a_{3} \cdot c_{3}, \\ a_{1} \cdot c_{2} + a_{2} \cdot c_{1} + a_{3} \cdot c_{2}, \\ a_{1} \cdot c_{3} + a_{3} \cdot c_{2} + a_{3} \cdot c_{1}) \\ - λ \cdot (c_{1} \cdot a_{1} - c_{2} \cdot a_{2} - c_{3} \cdot a_{3}, \\ c_{1} \cdot a_{2} + c_{2} \cdot a_{1} + c_{3} \cdot a_{2}, \\ c_{1} \cdot a_{3} + c_{3} \cdot a_{2} + c_{3} \cdot a_{1}) \\ = (0, λ \cdot a_{3} \cdot c_{2} - λ \cdot c_{3} \cdot a_{2}, \\ λ \cdot a_{3} \cdot c_{2} - λ \cdot c_{3} \cdot a_{2}) \\ = λ \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (0, 1, 1) . \end{matrix}

Second term:

γ \cdot [(b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] = γ \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (0, 1, 1) .

First term + second term:

\begin{matrix} λ \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (0, 1, 1) + γ \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (0, 1, 1) \\ = (0, λ \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}), λ \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2})) \\ + (0, γ \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}), γ \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2})) \\ = (0, λ \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) + γ \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}), \\ λ \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) + γ \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2})) \\ = (λ \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) + γ \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2})) \cdot (0, 1, 1) . \end{matrix}

Therefore:

\begin{matrix} (0, λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2} - c_{3} \cdot λ \cdot a_{2} - c_{3} \cdot γ \cdot b_{2}, \\ λ \cdot a_{3} \cdot c_{2} + γ \cdot b_{3} \cdot c_{2} - c_{3} \cdot λ \cdot a_{2} - c_{3} \cdot γ \cdot b_{2}) \\ = (0, λ \cdot a_{3} \cdot c_{2} - λ \cdot c_{3} \cdot a_{2} + γ \cdot b_{3} \cdot c_{2} - γ \cdot c_{3} \cdot b_{2}, \\ λ \cdot a_{3} \cdot c_{2} - λ \cdot c_{3} \cdot a_{2} + γ \cdot b_{3} \cdot c_{2} - γ \cdot c_{3} \cdot b_{2}) . \end{matrix}

Appendix E. Proof of the Jacobi Identity

The Jacobi identity is satisfied when:

\begin{matrix} [(a_{1}, a_{2}, a_{3}), [(b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})]] + \\ [(b_{1}, b_{2}, b_{3}), [(c_{1}, c_{2}, c_{3}), (a_{1}, a_{2}, a_{3})]] + \\ [(c_{1}, c_{2}, c_{3}), [(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})]] = \\ (0, 0, 0) . \end{matrix}

Proof:

\begin{matrix} [(b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})] = (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (0, 1, 1) = (0, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) . \end{matrix}

Appendix E.0.0.1. [(a 1 ,a 2 ,a 3 ),[(b 1 ,b 2 ,b 3 ),(c 1 ,c 2 ,c 3 )]]=(a 3 ·(b 3 ·c 2 -c 3 ·b 2 )-(b 3 ·c 2 -c 3 ·b 2 )·a 2 )·(0,1,1).

\begin{matrix} [(a_{1}, a_{2}, a_{3}), [(b_{1}, b_{2}, b_{3}), (c_{1}, c_{2}, c_{3})]] \\ = (a_{1}, a_{2}, a_{3}) ⊙ (0, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \\ - (0, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) ⊙ (a_{1}, a_{2}, a_{3}) . \end{matrix}

Appendix E.0.0.2. (a 1 ,a 2 ,a 3 )⊙(0,b 3 ·c 2 -c 3 ·b 2 ,b 3 ·c 2 -c 3 ·b 2 ):

\begin{matrix} (a_{1}, a_{2}, a_{3}) ⊙ (0, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \\ = (- a_{2} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) - a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}), \\ a_{1} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) + a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}), \\ a_{1} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) + a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2})) . \end{matrix}

Appendix E.0.0.3. (0,b 3 ·c 2 -c 3 ·b 2 ,b 3 ·c 2 -c 3 ·b 2 )⊙(a 1 ,a 2 ,a 3 ):

\begin{matrix} (0, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}, b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) ⊙ (a_{1}, a_{2}, a_{3}) \\ = (- (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2} - (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{3}, \\ (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{1} + (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2}, \\ (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2} + (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{1}) . \end{matrix}

\begin{matrix} (- a_{2} \cdot b_{3} \cdot c_{2} + a_{2} \cdot c_{3} \cdot b_{2} - a_{3} \cdot b_{3} \cdot c_{2} + a_{3} \cdot c_{3} \cdot b_{2}, \\ a_{1} \cdot b_{3} \cdot c_{2} - a_{1} \cdot c_{3} \cdot b_{2} + a_{3} \cdot b_{3} \cdot c_{2} - a_{3} \cdot c_{3} \cdot b_{2}, \\ a_{1} \cdot b_{3} \cdot c_{2} - a_{1} \cdot c_{3} \cdot b_{2} + a_{3} \cdot b_{3} \cdot c_{2} - a_{3} \cdot c_{3} \cdot b_{2}) \\ - (- b_{3} \cdot c_{2} \cdot a_{2} + c_{3} \cdot b_{2} \cdot a_{2} - b_{3} \cdot c_{2} \cdot a_{3} + c_{3} \cdot b_{2} \cdot a_{3}, \\ b_{3} \cdot c_{2} \cdot a_{1} - c_{3} \cdot b_{2} \cdot a_{1} + b_{3} \cdot c_{2} \cdot a_{2} - c_{3} \cdot b_{2} \cdot a_{2}, \\ b_{3} \cdot c_{2} \cdot a_{2} - c_{3} \cdot b_{2} \cdot a_{2} + b_{3} \cdot c_{2} \cdot a_{1} - c_{3} \cdot b_{2} \cdot a_{1}) \\ = (0, a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) - (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2}, \\ a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) - (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2}) \\ = (a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) - (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2}) \cdot (0, 1, 1) . \end{matrix}

Continuation of the proof:

\begin{matrix} [(b_{1}, b_{2}, b_{3}), [(c_{1}, c_{2}, c_{3}), (a_{1}, a_{2}, a_{3})]] \\ = (b_{3} \cdot (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) - (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot b_{2}) \cdot (0, 1, 1), \end{matrix}

\begin{matrix} [(c_{1}, c_{2}, c_{3}), [(a_{1}, a_{2}, a_{3}), (b_{1}, b_{2}, b_{3})]] \\ = (c_{3} \cdot (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) - (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot c_{2}) \cdot (0, 1, 1) . \end{matrix}

\begin{matrix} (a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) - (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2}) \cdot (0, 1, 1) \\ + (b_{3} \cdot (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) - (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot b_{2}) \cdot (0, 1, 1) \\ + (c_{3} \cdot (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) - (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot c_{2}) \cdot (0, 1, 1) \\ = (a_{3} \cdot (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) - (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot a_{2}) \cdot (0, 1, 1) \\ + (b_{3} \cdot (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) - (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot b_{2}) \cdot (0, 1, 1) \\ + (c_{3} \cdot (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) - (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot c_{2}) \cdot (0, 1, 1) \\ = ((a_{3} \cdot b_{3} \cdot c_{2} - a_{3} \cdot c_{3} \cdot b_{2}) - (b_{3} \cdot c_{2} \cdot a_{2} - c_{3} \cdot b_{2} \cdot a_{2})) \cdot (0, 1, 1) \\ + ((b_{3} \cdot c_{3} \cdot a_{2} - b_{3} \cdot a_{3} \cdot c_{2}) - (c_{3} \cdot a_{2} \cdot b_{2} - a_{3} \cdot c_{2} \cdot b_{2})) \cdot (0, 1, 1) \\ + ((c_{3} \cdot a_{3} \cdot b_{2} - c_{3} \cdot b_{3} \cdot a_{2}) - (a_{3} \cdot b_{2} \cdot c_{2} - b_{3} \cdot a_{2} \cdot c_{2})) \cdot (0, 1, 1) \\ = (a_{3} \cdot b_{3} \cdot c_{2} - a_{3} \cdot c_{3} \cdot b_{2} - b_{3} \cdot c_{2} \cdot a_{2} + c_{3} \cdot b_{2} \cdot a_{2} \\ + b_{3} \cdot c_{3} \cdot a_{2} - b_{3} \cdot a_{3} \cdot c_{2} - c_{3} \cdot a_{2} \cdot b_{2} + a_{3} \cdot c_{2} \cdot b_{2} \\ + c_{3} \cdot a_{3} \cdot b_{2} - c_{3} \cdot b_{3} \cdot a_{2} - a_{3} \cdot b_{2} \cdot c_{2} + b_{3} \cdot a_{2} \cdot c_{2}) \cdot (0, 1, 1) \\ = (0, 0, 0) . \end{matrix}

Appendix F. Proof of the Malcev Identity

The Malcev identity holds if:

\forall a, b, c \in M : [[a, b], [a, c]] = [[[a, b], c], a] + [[[b, c], a], a] + [[[c, a], a], b] .

Defining these

a, b, c

a = (a_{1}, a_{2}, a_{3})

b = (b_{1}, b_{2}, b_{3})

, and

c = (c_{1}, c_{2}, c_{3})

, we have:

[a, b] = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (0, 1, 1) .

[a, c] = (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (0, 1, 1) .

\begin{matrix} [[a, b], [a, c]] \\ = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (0, 1, 1) ⊙ (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (0, 1, 1) \\ - (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (0, 1, 1) ⊙ (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (0, 1, 1) \\ = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (0, 1, 1) ⊙ (0, 1, 1) \\ - (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (0, 1, 1) ⊙ (0, 1, 1) \\ = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (- 1 - 1, 1, 1) \\ - (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (- 1 - 1, 1, 1) \\ = ((a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \\ - (a_{3} \cdot c_{2} - c_{3} \cdot a_{2}) \cdot (a_{3} \cdot b_{2} - b_{3} \cdot a_{2})) \cdot (- 1 - 1, 1, 1) \\ = (0, 0, 0) . \end{matrix}

[[a, b], c] = [(a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (c_{2} - c_{3}) \cdot (0, 1, 1)] .

\begin{matrix} [[[a, b], c], a] \\ = [(a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (c_{2} - c_{3}) \cdot (0, 1, 1) ⊙ (a_{1}, a_{2}, a_{3}) \\ - (a_{1}, a_{2}, a_{3}) ⊙ (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (c_{2} - c_{3}) \cdot (0, 1, 1)] \\ = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (c_{2} - c_{3}) \cdot \\ [((- a_{2} - a_{3}, a_{1} + a_{2}, a_{2} + a_{1}) - (- a_{2} - a_{3}, a_{1} + a_{3}, a_{1} + a_{3}))] \\ = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (c_{2} - c_{3}) \cdot (0, a_{2} - a_{3}, a_{2} - a_{3}) \\ = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (c_{2} - c_{3}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) . \end{matrix}

[[b, c], a] = (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) .

\begin{matrix} [[[b, c], a], a] \\ = (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) ⊙ (a_{1}, a_{2}, a_{3}) \\ - (a_{1}, a_{2}, a_{3}) ⊙ (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) \\ = (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (a_{2} - a_{3}) \\ \times [((- a_{2} - a_{3}, a_{1} + a_{2}, a_{2} + a_{1}) - (- a_{2} - a_{3}, a_{1} + a_{3}, a_{1} + a_{3}))] \\ = (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (a_{2} - a_{3}) \cdot (0, a_{2} - a_{3}, a_{2} - a_{3}) \\ = (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (a_{2} - a_{3}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) . \end{matrix}

[[c, a], a] = (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) .

\begin{matrix} [[[c, a], a], b] \\ = (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) ⊙ (b_{1}, b_{2}, b_{3}) \\ - (b_{1}, b_{2}, b_{3}) ⊙ (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) \\ = (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot (a_{2} - a_{3}) \\ \times [((- b_{2} - b_{3}, b_{1} + b_{2}, b_{2} + b_{1}) - (- b_{2} - b_{3}, b_{1} + b_{3}, b_{1} + b_{3}))] \\ = (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot (a_{2} - a_{3}) \cdot (b_{2} - b_{3}) \cdot (0, 1, 1) . \end{matrix}

\begin{matrix} [[[a, b], c], a] + [[[b, c], a], a] + [[[c, a], a], b] \\ = (a_{3} \cdot b_{2} - b_{3} \cdot a_{2}) \cdot (c_{2} - c_{3}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) \\ + (b_{3} \cdot c_{2} - c_{3} \cdot b_{2}) \cdot (a_{2} - a_{3}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) \\ + (c_{3} \cdot a_{2} - a_{3} \cdot c_{2}) \cdot (a_{2} - a_{3}) \cdot (b_{2} - b_{3}) \cdot (0, 1, 1) \\ = (a_{3} \cdot b_{2} \cdot c_{2} - a_{3} \cdot b_{2} \cdot c_{3} - b_{3} \cdot a_{2} \cdot c_{2} + b_{3} \cdot a_{2} \cdot c_{3} \\ + b_{3} \cdot c_{2} \cdot a_{2} - b_{3} \cdot c_{2} \cdot a_{3} - c_{3} \cdot b_{2} \cdot a_{2} + c_{3} \cdot b_{2} \cdot a_{3} \\ + c_{3} \cdot a_{2} \cdot b_{2} - c_{3} \cdot a_{2} \cdot b_{3} - a_{3} \cdot c_{2} \cdot b_{2} + a_{3} \cdot c_{2} \cdot b_{3}) \cdot (a_{2} - a_{3}) \cdot (0, 1, 1) \\ = (0, 0, 0) . \end{matrix}

[[a, b], [a, c]] = [[[a, b], c], a] + [[[b, c], a], a] + [[[c, a], a], b] = (0, 0, 0) .

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First Exploration of a Novel Generalization of Lie and Malcev Algebras Leading to the Emergence of Complex Structures

Abstract

1. Introduction

1.1. Lie Algebra

1.2. Malcev Algebras

2. Construction of Algebra ( V , ⊙ )

2.1. Algebraic Structure of the Vector Space V

2.2. Definition of the Algebra ( V , ⊙ )

2.2.1. Bilinearity of the Product

2.2.2. Basic Properties of Algebra B

2.2.2.1. Non-Associativity and Non-Alternativity

2.2.2.2. Non-Commutativity and Antisymmetry

2.2.2.3. Identity Element

2.3. Relation of Algebra B with Malcev and Lie Algebras

2.3.1. Relation of Algebra B with Lie Algebra

2.3.1.1. Bilinearity of the Commutator

2.3.1.2. Jacobi Identity

2.3.2. Construction of Malcev Algebra through Algebra B

3. Generation of Complex Entities from B

3.1. Imaginary Unit i in B

3.2. Imaginary Unit j in B

4. Other Properties

4.1. Nullification of the Product a · i ⊙ b · j

4.2. Product a 3 · j ⊙ b 2 · i

5. Conclusions

Appendix A. Proof of the Bilinearity of the Product ⊙

Appendix B. Proof of the Non-Associativity of the Product ⊙

Appendix C. Proof of the Non-Alternativity of the Product ⊙

Appendix D. Proof of the bilinearity of the Lie bracket defined from ⊙

Appendix E. Proof of the Jacobi Identity

Appendix E.0.0.1. [(a 1 ,a 2 ,a 3 ),[(b 1 ,b 2 ,b 3 ),(c 1 ,c 2 ,c 3 )]]=(a 3 ·(b 3 ·c 2 -c 3 ·b 2 )-(b 3 ·c 2 -c 3 ·b 2 )·a 2 )·(0,1,1).

Appendix E.0.0.2. (a 1 ,a 2 ,a 3 )⊙(0,b 3 ·c 2 -c 3 ·b 2 ,b 3 ·c 2 -c 3 ·b 2 ):

Appendix E.0.0.3. (0,b 3 ·c 2 -c 3 ·b 2 ,b 3 ·c 2 -c 3 ·b 2 )⊙(a 1 ,a 2 ,a 3 ):

Appendix F. Proof of the Malcev Identity

References

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2. Construction of Algebra $(V, ⊙)$

2.2. Definition of the Algebra $(V, ⊙)$

4.1. Nullification of the Product $a \cdot i ⊙ b \cdot j$

4.2. Product $a_{3} \cdot j ⊙ b_{2} \cdot i$