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We study real algebras admitting reflections which commute. In dimension two, we show that two commuting reflections coincide. We specify the two and four-dimensional real algebras cases. We characterize real algebras of division of two-dimensional to third power associative having a reflection. Finally We give a characterization in four-dimensional, the unitary real algebras of division at third power-associative having two reflections that commute. In eight-dimensional, we give an example of algebra so the group of automorphisms contains a subgroup isomorphic to \(\mathbb{Z}_2\times\mathbb{Z}_2\).
The classification of non-associative division algebras over a commutative field \(\mathbb{K}\) with a characteristic different from \(2\) is a pastionning and topical problem, whose origins date back to the discovery of quaternions (\(\mathbb{H}\), Hamilton 1843) and octonions (\(\mathbb{O}\), Graves 1843, Cayley 1845). Fundamental results appeared, Hopf proved that the dimension of a real algebra of division of finite dimension \(n\) is a power of 2 and cannot exceed 2 in the commutative case.
Bott and Milnor [1] refined the result of Hopf by reducing the power of \(2\) to \(n\in \{1,2, 4, 8\}\). It is trivial to show that in dimension one the real algebra \(\mathbb{R}\) is unique. In two-dimensional, the classification of these algebras is recently completed. The problem remains open in dimension 4 and 8. The study is quite interesting when there is a sufficient number of distinct reflections which commute for a finite-dimensional division algebra since the product of this last translates a certain symmetry and elegance.
In this paper, we give a description of real algebras of division in two-dimensional having one reflection, and in four-dimensional having two distinct reflections which commute. We recall that the subgroup generated by the latter is isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\) with \(\displaystyle\mathbb{Z}_2:=\frac{\mathbb{Z}}{2\mathbb{Z}}\).
In eight-dimensional, we give a class of algebra whose group of automorphisms contains the Klein’s group without giving the necessary and sufficient condition of the division.
Let \(A\) be an arbitrary non-associative real algebra. We define, \(I(A)=\{x\in A, x^2=x\}\) and let \(x\), \(y\in A\), \([x,y]=xy-yx\) and \((x,y,z)=(xy)z-x(yz)\).
\(A\) is said to be
– division if the operations \(L_x: A\rightarrow A\), \(y\mapsto xy\) and \(R_x: A\rightarrow A\), \(y\mapsto yx\) are bijective, for all \(x\in A\), \(x\neq 0.\)
–at third power-associative if \((x,x,x)=0\) for all \(x\in A\).
–at 121 power-associative if \((x,x^2,x)=0\) for all \(x\in A\).
A linear map \(\partial: A\rightarrow A\) is said to be a derivation of \(A\) if for all \(x,\) \(y\in A\), we have \(\partial(x.y)=\partial(x).y+x.\partial(y)\). The derivations of \(A\) form a vector subspace of the endomorphisms of \(A\) (\(End_\mathbb{K}(A)\)) which is a Lie algebra, for the bracket of Lie \([f,g]=f\circ g- g\circ f\). Such an algebra is called the Lie algebra of the derivations of \(A\) denoted by \(Der(A).\)
A linear map \(f: A\rightarrow A\) is said to be an automorphism of \(A\) if \(f\) is bijective and for all \(x,\) \(y\in A\), \(f(x.y)=f(x).f(y).\) The automorphisms of \(A\) constitute a group \(Aut_\mathbb{K}(A)\) for the usual law. \(f\in Aut(A)\) is said to be a reflection of \(A,\) if \(f\) is involutive \((f\circ f=id_A)\) not identical \((f\neq id_A)\). Let \(f\) be an automorphism of \(A\) and \(\lambda \in \mathbb{R}\), we denote by \(E_{\lambda}(f)\) the kernel of \(f-\lambda id_A\), \(id_A\) being the identity operator of \(A\).
Let \(f, g: A\rightarrow A\) be linear bijections of \(A\). Recall that the \((f,g)\)-Albert isotope of \(A\) denoted by \(A_{f,g}\) is a vector space \(A\) with the product \(x\odot y = f(x)g(y)\).
Remark 1. A linearization of \((x,x,x)=0\), gives \([x^2,y]+[xy+yx,x]=0\) for all \(x\), \(y\in A\). So by taking \(y=x^2\) we get \([x.x^2+x^2.x, x]=0\Rightarrow 2[x.x^2,x]=0\Rightarrow (x.x^2)x-x(x.x^2)=0\Rightarrow (x.x^2)x-x(x^2.x)=0\Rightarrow (x,x^2,x)=0\). We can therefore affirm that if \(A\) is at third power-associative then it is at 121 power-associative. But the converse is not true in the case where the algebra is not of division. For example the algebra \(A\) having a basis \(\{e_1, e_2, e_3, e_4\}\) whose product of the elements in this base is given by, \({e_1}^2=e_1\), \(e_1e_2=e_3\), \(e_2e_1=e_4\) and other null products, is at 121 power-associative and it is not at third power-associative.
In [2], we have the following result;
Lemma 1. Let \(A\) be a real algebra of division of finite 2n-dimensional with \(n\in\{1,2,4\}\). We suppose that there exists an automorphism \(f\) of \(A\) such that \(sp(f)=\{-1,1\}\). Then the following inclusions between subalgebras of \(A\) are strict, \[\{0\}\subset E_1(f)\subset E_1(f)+E_{-1}(f).\] In that case \(dim E_1(f)=n\geq 1,\) and \(dim \big( E_1(f)+E_{-1}(f)\big)=2n.\)
In addition, the following statements are equivalent,
And the following equalities hold, \[E_1(f)E_{-1}(f)=E_{-1}(f)E_1(f)=E_{-1}(f),\] \[E_{-1}(f).E_{-1}(f)=E_1(f).\]
Definition 1. Let \(\alpha\), \(\beta,\) \(\gamma,\) \(\alpha',\) \(\beta'\) and \(\gamma'\) be real numbers. We define
algebra \(A\) having a base \(\{e,e_1, e_2, e_3\}\) for which the product
is given as
\[
\label{tab1}\tag{1}
\begin{array}{c|c|c|c|c}
\odot & e & e_1 & e_2 & e_3\\
\hline
e & e & e_1 & e_2 & e_3\\
\hline
e_1 & e_1 & -e & \gamma e_3 & \beta’ e_2\\
\hline
e_2 & e_2 & \gamma’ e_3 & -e & \alpha e_1\\
\hline
e_3 & e_3 & \beta e_2 & \alpha’ e_1 & -e\\
\end{array}
\]
We denote this algebra \(A\) by \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\).
In [3], the author gives the following results:
Theorem 1. A necessary condition that the algebra \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) should be a division algebra, is that the six nonzero constants \(\alpha,\) \(\beta,\) \(\gamma,\) \(-\alpha',\) \(-\beta',\) \(-\gamma'\) should have the same sign. This sign may be taken to be positive without loss of generality. If the six constants are positive, a sufficient (but not necessary) condition for a division algebra is that they satisfy the relation \(f(\alpha, \beta, \gamma)=f(-\alpha', -\beta', -\gamma')\) where \(f\) is defined by \[f(x,y,z)=x+y+z-xyz.\]
Theorem 2. A necessary and sufficient condition that the division algebra \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) should have two sided rank 2 is that the equations \(\alpha'=\alpha\), \(\beta'=\beta\) and \(\gamma'=\gamma\) hold true. In the contrary case, the algebra has two sided rank 4.
Proposition 1. Let \(A\) be a finite dimensional real algebra. The following proposals are equivalent,
\(Aut(A)\) contains two distinct reflections which commute.
\(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
Proof. \((1)\Longrightarrow (2)\) Let \(A\) be a real algebra such that \(Aut(A)\) contains two reflections which commute \(f\) and \(g.\) Then \(h=f\circ g \in Aut(A)\) and is a different reflection of \(f\) and \(g\). The subgroup of \(Aut(A)\) generated by \(f\) and \(g\) is isomorphic to the klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
\((2)\Longrightarrow (1)\) Obvious because the elements of the subgroup of \(Aut(A)\) which is isomorphic to \(\mathbb{Z}_2\times \mathbb{Z}_2\) different from the identity are distinct reflections that commute. ◻
Proposition 2. Let \(A\) be a two-dimensional division algebra and let \(f\), \(g\) be two commuting reflections of \(A\). Then \(f=g\)
Proof. The endomorphisms \(f\) and \(g\) of \(A\) are diagonalizable which commute, consequently there exists a common basis \(\{e, e_1\}\) of \(A\), formed of eigenvectors associated with eigenvalues \(1\) and \(-1\). According to Lemma 1. The eigpaces \(E_1(f)\), \(E_{-1}(f)\), \(E_1(g)\), \(E_{-1}(g)\) of \(f\) and \(g\) are one-dimensional. By setting \(E_1(f)= \mathbb{R}e\) and \(E_{-1}(f)=\mathbb{R}e_1\).
Suppose that \(E_1(g)=\mathbb{R}e_1\) and \(E_{-1}(g)=\mathbb{R}e\), let \(x=a e+b e_1\in A,\) we have \[f(x)=af(e)+bf(e_1)=ae-be_1=-ag(e)-bg(e_1)=-g(ae+be_1)=-g(x),\] thus \(f=-g\) (which is not an automophism of \(A\)), absurd. So \(E_1(g)=\mathbb{R}e\) and \(E_{-1}(g)=\mathbb{R}e_1\), therefore \(f\) and \(g\) coincide on \(A=E_1(f)\oplus E_{-1}(f).\) ◻
Theorem 3. Let \(A\) be a two-dimensional division algebra
and let \(f\) be a reflections of \(A.\) Then there exists a basis \(B=\{e, e_1\}\) of \(A\) such that the product of \(A\) in this basis is given as,
\[
\label{tab2}\tag{2}
\begin{array}{c|c|c}
\odot & e & e_1 \\
\hline
e & e & \alpha e_1 \\
\hline
e_1 & \beta e_1 & \gamma e \\
\end{array}
\]
with \(\alpha\),
\(\beta\), \(\gamma\) are non-zero real numbers and
\(\alpha\beta\gamma< 0\). We will
denote this algebra \(A(\alpha, \beta,
\gamma)\).
Proof. The eigenspaces \(E_1(f)\), \(E_{-1}(f)\) of f are one-dimensional and \(A=E_1(f) \oplus E_{-1}(f)\). There exists \(e \in E_1(f)\) and \(e_1 \in E_{-1}(f)\), such that \(\{e, e_1\}\) is a basis of \(A.\) Taking into account the equalities \((\ast)\) of Lemma 1, we obtain the product of the elements of this basis of (2). ◻
For the division, the Theorem 3 of [4] gives the result.
Corollary 1. Let \(A\) be a two-dimensional division algebra, then the following propositions are equivalent,
\(Aut(A)\) contains a reflection.
\(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha\beta\gamma< 0\).
Proof. \((1)\Rightarrow (2)\) The Theorem 3 gives the result.
\((2) \Rightarrow (1)\) The endomorphism \(f\) of \(A(\alpha, \beta, \gamma)\) defined by \(f(x_0e+x_1e_1)=x_0e-x_1e_1\) is a reflection of \(A(\alpha, \beta, \gamma)\). ◻
Theorem 4. Let \(A\) be a two-dimensional division algebra and let \(f\) be a reflection of \(A\), then the following propositions are equivalent,
\(A\) is commutative.
\(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\beta=\alpha\) and \(\gamma< 0\).
\(A\) is at third power-associative.
\(A\) is at 121 power-associative.
Proof. \((1)\Rightarrow (2)\) Since \(f\) be a reflection of \(A\), the Corollary 1 asserts that \(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha \beta \gamma< 0\). Since \(A\) is commutative, \(e_1e=ee_1\Longrightarrow \alpha=\beta\), therefore we get the result.
\((2)\Rightarrow (3)\) It is easy to show that \(A(\alpha, \beta, \gamma)\), with \(\beta=\alpha\) and \(\gamma< 0\), is at third power-associative.
\((3)\Rightarrow (4)\) The Remark 1 gives the result.
\((4)\Rightarrow (1)\) \(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha \beta \gamma< 0\) and it is at \(121\) power-associative, We have \((e_1, {e_1}^2, e_1)=0\Rightarrow \gamma^2(\alpha-\beta)e=0\Rightarrow \alpha=\beta.\) and \(\gamma< 0.\) Then \(A\) is commutative. ◻
Theorem 5. Let \(A\) be a two-dimensional division algebra
and let \(f\) be a reflection of \(A\). We are obtained the idempotents and
the derivations of \(A.\)
\[
\begin{array}{c|c|c}
& I\big(A(\alpha, \beta, \gamma)\big) & Der(A(\alpha, \beta, \gamma)) \\
\hline
\alpha+\beta\neq 0 \text{ et } \gamma(\alpha+\beta-1) > 0 & \{e, \lambda_0e-\lambda_1 e_1, \lambda_0e+\lambda_1 e_1 \} & \{0\} \\
\hline
\text{otherwise} & \{e\} & \{0\} \\
\end{array}
\]
with \(\lambda_0=\frac{1}{\alpha+\beta}\), \({\lambda_1}^2=\frac{\alpha+\beta-1}{\gamma(\alpha+\beta)^2}\).
Proof. \(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha\beta\gamma< 0.\)
Let \(x=\lambda_0 e+\lambda_1e_1 \in I(A)\), we have \[x^2=x\Leftrightarrow \left\{\begin{array}{ll} {\lambda_0}^2+\gamma{\lambda_1}^2= \lambda_0\\ (\alpha+\beta)\lambda_0\lambda_1=\lambda_1 \end{array}\right.\] We obtain \(I(A)\) by resolving the system and discussing on \(\alpha+\beta\) and \(\displaystyle \frac{\alpha+\beta-1}{\gamma}\).
\(A(\alpha, \beta, \gamma)\) with \(\alpha\beta\gamma< 0\) be a two-dimensional division algebra, then \(Der(A)=\{0\}.\)
◻
Proposition 3. Let \(A\) be a two-dimensional division algebra and \(f\) be a reflection of \(A\). Then the following propositions are equivalent,
\(Aut(A)\) is isomorphic to \(S_3\).
\(A\) is isomorphic to McClay \(\stackrel{\ast}{\mathbb{C}}:=(\mathbb{C}, \odot)\) with \(x\odot y=\overline{x} \ \overline{y}\) for \(x\), \(y\in \mathbb{C}\).
Here, \(\overline{x}\) and \(\overline{y}\), are the respective conjugates of \(x\) and \(y\).
Proof. \((1)\Longrightarrow (2)\) By hypothesis \(A\) is isomorphic to \(A(\alpha, \beta, \gamma)\) with \(\alpha\beta\gamma< 0\) and \(Aut(A)\simeq S_3\), then there exists \(f\in Aut(A)\) not identical of order 3.
We suppose that \(\alpha+\beta=0\) where \(\gamma(\alpha+\beta-1)\leq 0\). Since \(f(e)\in I(A)=\{e\}\) then \(f(e)=e\) and \(f(e_1)=\pm e_1\) therefore \(f^2=id_A\), absurd because \(f\) is of order 3. We now assume that \(\alpha+\beta\neq 0\) and \(\gamma(\alpha+\beta-1)> 0\). Since \(f(e)\in I(A)=\{e, \lambda_0 e_0+\lambda_1 e_1, \lambda_0 e_0-\lambda_1 e_1\}\). If \(f(e)=e\) we have \(f(e_1)=\pm e_1\), contradiction because \(f\) is of order 3. There is only one case left \(f(e)=\lambda_0 e\pm\lambda_1 e_1\). Let \(f(e_1)=x_0e_0+x_1e_1\), with \(x_0, x_1\) non-zero real numbers.
Looking at the components of \(e\) of the equations \(f(e)f(e_1)=\alpha f(e_1)\) et \(f(e_1)f(e)=\beta f(e_1)\) we have \(\alpha x_0=\beta x_o \Rightarrow \alpha=\beta\). Therefore \(A\) is commutative so it is reflexive. In conclusion \(A\) is reflexive and \(Aut(A)\simeq S_3\), hence the result follows.
\((2)\Longrightarrow (1)\) By simple calculation we have \(Aut(\stackrel{\ast}{\mathbb{C}})\simeq S_3.\) ◻
Proposition 4. Let \(A\) be a four-dimensional division algebra, let \(f\) and \(g\) are reflections of \(A.\) Then there exists a basis \(B=\{e, e_1, e_2, e_3\}\) of \(A\) where \(e^2=e\) and \[E_1(f)=\mathbb{R}e+\mathbb{R}e_1,\] \[E_{-1}(f)=\mathbb{R}e_2+\mathbb{R}e_3,\] \[E_1(g)=\mathbb{R}e+\mathbb{R}e_2,\] and \[E_{-1}(g)=\mathbb{R}e_1+\mathbb{R}e_3.\]
Proof. Since \(f\) and \(g\) are diagonalizable and commute, so there is a common basis \(\{e,e_1,e_2, e_3\}\) formed by eigenvectors associated to eigenvalues \(1\) or \(-1\). The Lemma 1 shows that \(E_1(f)\), \(E_{-1}(f)\), \(E_1(g)\) and \(E_{-1}(g)\) are vector spaces of two-dimensional. If \(E_1(f)=\mathbb{R}e+\mathbb{R}e_1\), \(E_{-1}(f)=\mathbb{R}e_2+\mathbb{R}e_3\), then one of the eigenvectors \(e\), \(e_1\) (and only one) belongs to \(E_{1}(g)\), otherwise \(f\) and \(g\) coincide in \(A=E_1(f)\oplus E_{-1}(f)\).
We can therefore set \(E_1(g)=\mathbb{R}e+\mathbb{R}e_2\) and \(E_{-1}(g)=\mathbb{R}e_1+\mathbb{R}e_2\). We have \[e_1\in E_1(f)\Rightarrow {e_1}^2\in E_1(f).E_1(f)=E_1(f),\] and \[e_1\in E_{-1}(g)\Rightarrow {e_1}^2\in E_{-1}(g).E_{-1}(g)=E_1(g).\] Therefore \(e_1\in E_1(f)\cap E_1(g)=\mathbb{R}e\). Thus by analogy, the elements \({e_i}^2\in E_1(f)\cap E_1(g)=\mathbb{R}e\) for all \(i\in \{2, 3\}\). Since \(E_1(f)\cap E_1(g)\) is a subalgebra of \(A\), the element \(e\) is a scalar multiple of an idempotent and can be assumed to be idempotent. ◻
Proposition 5. Let \(A\) be a real division algebra of unit four-dimensional of unit \(e\). Then the following propositions are equivalent,
\(Aut(A)\) contains two distinct reflections which commute.
\(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
\(A\) is isomorphic to the algebra \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\).
Proof. \((1) \Longleftrightarrow (2)\) This is true according to Proposition 1.
\((2) \Longleftrightarrow (3)\) We have two distinct reflections which commute. Proposition 2 ensures the existence of a basis \(\{e, e_1, e_2, e_3\}\) of \(A\), and the subalgebras \(\mathbb{R}e+\mathbb{R}e_i\), for all \(i\in \{1,2,3\}\), are isomorphic to the algebra \(\mathbb{C}\) since they are real algebras of unit division of two-dimensional. We can now set \[E_1(f)=\mathbb{R}e+\mathbb{R}e_1,\] \[E_{-1}(f)=\mathbb{R}e_2+\mathbb{R}e_3,\] \[E_1(g)=\mathbb{R}e+\mathbb{R}e_2\] and \[E_{-1}(g)=\mathbb{R}e_1+\mathbb{R}e_3.\]
We have \(e_1\in E_1(f) \ et \ e_2 \in E_{-1}(f)\Rightarrow e_1e_2, \ \ e_2e_1\in E_1(f).E_{-1}(f)=E_{-1}(f)\), \(e_1 \in E_{-1}(g)\) and \(e_2 \in E_1(g)\Rightarrow e_1e_2, \ \ e_2e_1 \in E_{-1}(g).E_1(g)=E_{-1}(g)\).
As a result \(e_1e_2,\) and \(e_2e_1 \in E_{-1}(f)\cap E_{-1}(g)=\mathbb{R}e_3\). In the same way we have \[\begin{aligned} e_1e_3, && e_3 e_1 \in E_{-1}(f)\cap E_{1}(g)=\mathbb{R}e_2, \\ e_2e_3, && e_3 e_2 \in E_{1}(f)\cap E_{-1}(g)=\mathbb{R}e_1. \end{aligned}\] This gives the multiplication table for (1).
\((3)\Longrightarrow (1)\) \(f: A\longrightarrow A;\) \(\lambda_0 e+\sum\limits_{i=1}^{3}\lambda_i e_i\longmapsto \lambda_0 e+\lambda_1 e_1-\sum\limits_{i=2}^{3}\lambda_i e_i\) and \(g: A\longrightarrow A;\) \(\lambda_0 e+\sum\limits_{i=1}^{3}\lambda_i e_i\longmapsto \lambda_0 e-\lambda_1 e_1+\lambda_2 e_2-\lambda_3e_3\) are distinct reflections that commute. ◻
Corollary 2. If \(A\) is a real division algebra of four-dimensional with two commute distinct reflections then \(A\) is isotope in the sens of Albert to \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\) with \(e\in I(A)\).
Proof. Let \(A\) be a real division algebra of four-dimensional having two distinct reflections that commute \(f\) and \(g\). According to Proposition 2 there is a basis \(\{e, e_1, e_2, e_3\}\) where \(e\in I(A)\) and \(f(e)=g(e)=e\). Let \(x \in A\), we have \[f\circ R_e(x)=f(R_e(x))=f(xe)=f(x)f(e)=f(x)e=R_e(f(x))=R_e\circ f(x),\] so \[f\circ R_e=R_e\circ f\Leftrightarrow f\circ {R_e}^{-1}={R_e}^{-1}\circ f.\] As well as \(f\circ {L_e}^{-1}={L_e}^{-1}\circ f\), \((A, \odot)\) where \(x\odot y={R_e}^{-1}(x).{L_e}^{-1}(y)\) for all \(x\), \(y \in A\), is isotope to \(A\), and is unitary of unit \(e\). We have, \[f(x\odot y)=f\big({R_e}^{-1}(x).{L_e}^{-1}(y)\big)=f\big({R_e}^{-1}(x)\big).f\big({L_e}^{-1}(y)\big)={R_e}^{-1}(f(x)).{L_e}^{-1}(f(y))=f(x)\odot f(y),\] then \(f\in Aut((A,\odot))\). We also show by analogy that \(g\in Aut((A,\odot)).\) Hence \((A,\odot)\) verifies the assumptions of Proposition 5 therefore it is isomorphic to \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\). ◻
Theorem 6. Let \(A\) be a real division algebra of unit four-dimensional having two reflections that commute. Then the following statements are equivalent,
\(A\) is isomorphic to \(\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\), with \(\alpha'=\alpha\), \(\beta'=\beta\), \(\gamma'=\gamma\).
\(A\) is at third power-associative.
\(A\) is at 121 power-associative.
Proof. \((1)\Longrightarrow(2)\) For all \(x=\displaystyle \lambda_0e+\sum\limits_{i=1}^{3}\lambda_ie_i\in\mathbb{H}(\alpha, \beta, \gamma, \alpha', \beta', \gamma')\), with \(\alpha'=\alpha\), \(\beta'=\beta\), \(\gamma'=\gamma\), we have \(x^2=-N(x)e+2\lambda_0 x\) with \(N(x)=\displaystyle\sum\limits_{i=0}^{3}{\lambda_i}^2\) and it is easy to check that \((x,x,x)=0\)
\((2)\Longrightarrow(3)\) It is clear from Remark 1
\((3)\Longrightarrow(1)\) A simple calculation show that \(\alpha=\alpha'\), \(\beta=\beta'\) et \(\gamma=\gamma'\). ◻
Proposition 6. Let \(A\) be a division algebra of eight-dimensional, the following statements are equivalent,
\(Aut(A)\) contains two distinct reflections which commute.
\(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
There are vector subspaces \(X\), \(Y,\) \(Z\)
and \(T\) of two-dimensional, for which
the multiplication of \(A\) is given
by,
\[
\label{tab3}\tag{3}
\begin{array}{c|c|c|c|c}
& X & Y & Z & T \\
\hline
X & X & Y & Z & T \\
\hline
Y & Y & X & T & Z \\
\hline
Z & Z & T & X & Y \\
\hline
T & T & Z & Y & X \\
\end{array}
\]
Proof. \((1)\Longrightarrow (2)\) This is true according to Proposition 5.
\((2)\Longrightarrow (3)\) The group \(Aut(A)\) contains two distinct reflections \(f\) and \(g\) which commute. Then there is a basis \(\{e, e_1, e_2, e_3, e_4, e_5, e_6, e_7\}\) consisting of eigenvectors common to \(f\) and \(g\) where \(E_1(f)=Lin\{e, e_1, e_2, e_3\}\) and \(E_{-1}(f)=Lin\{e_4, e_5, e_6, e_7\}\). As \(f\neq g\) the subalgebras \(E_1(f)\), \(E_1(g)\) of dimension \(4\), cannot coincide. Consequently the subalgebra \(X:=E_1(f)\cap E_1(g)\), is of dimension \(\leq 2.\)
Moreover the subalgebra \(X\) cannot be reduced to \(\mathbb{R}e.\) Otherwise the vector subspace \(Lin\{e_1,e_2, e_3\}:=E\) of \(E_1(f)\), would be contained in \(E_{-1}(g)\) and we would have \(E^2\subset E_1(f)^2 \cap E_{-1}(g)^2= E_1(f)\cap E_1(g)=\mathbb{R}e\) nonsense because, \(e_1\), \(e_2\) \(e_3\in E\) then \(e_1e_2, e_1e_3\in E^2\subset \mathbb{R}e\) so there exist \(\alpha, \beta \in \mathbb{R}\) nonzero, such that \(e_1.e_2=\alpha e\) and \(e_1e_3=\beta e\). We have \(e_1.(\beta e_2-\alpha e_3)=0\Longrightarrow \beta e_2-\alpha e_3=0\), as \(A\) is of division so \(\beta e_2=\alpha e_3\) contradicting the fact that \(e_2\) and \(e_3\) are linearly independent. So \(X\) is of dimension 2.
If for example \(X=Lin\{e, e_1\}\) we can state that \(E_1(g)=Lin\{e, e_1, e_4, e_5\}\) and \(E_{-1}(g)=Lin\{e_2, e_3, e_6, e_7\}\), we then obtain the following sub-vector spaces of dimension \(2\).
\(Y:=E_1(f)\cap E_{-1}(g)=Lin\{e_2,
e_3\}\),
\(Z:=E_1(g)\cap E_{-1}(f)=Lin\{e_4,
e_5\}\),
\(T:=E_{-1}(f)\cap E_{-1}(g)=Lin\{e_6,
e_7\}\).
It is easy to show that the multiplication of \(A\) is done according to (3).
\((3)\Longrightarrow (1)\) The vector space \(A\) decomposes into a direct sum of the subspaces vector espaces \(X\), \(Y\), \(Z\), \(T\) and the two endomorphisms \(f, g: A:=X\oplus Y\oplus Z\oplus T\rightarrow X\oplus Y\oplus Z\oplus T\) defined by, for all \(u=x+y+z+t\in X\oplus Y\oplus Z\oplus T\) we have \(f(u)= x+y-z-t\) and \(g(u)= x-y+z-t\). They are distinct reflections, which commute. Thus the subgroup of \(Aut(A)\) generated by \(f\) and \(g\) is isomorphic to \(\mathbb{Z}_2\times \mathbb{Z}_2\). ◻
Example 1. Let \(A\) be a division algebra of eight-dimensional whose product in the base \(B=\{e, u_1, \ldots, u_7\}\) is given by,
| . | e | \(u_1\) | \(u_2\) | \(u_3\) | \(u_4\) | \(u_5\) | \(u_6\) | \(u_7\) |
|---|---|---|---|---|---|---|---|---|
| e | e | \(u_1\) | \(u_2\) | \(u_3\) | \(u_4\) | \(u_5\) | \(u_6\) | \(u_7\) |
| \(u_1\) | \(u_1\) | \(-e\) | \(\gamma u_3\) | \(-\beta' u_2\) | \(\delta u_5\) | \(-\eta'u_4\) | \(\lambda u_7\) | \(-\mu'u_6\) |
| \(u_2\) | \(u_2\) | \(-\gamma'u_3\) | \(-e\) | \(\alpha u_1\) | \(\sigma u_6\) | \(\eta u_7\) | \(-\rho'u_4\) | \(-\xi'u_5\) |
| \(u_3\) | \(u_3\) | \(\beta u_2\) | \(-\alpha'u_1\) | \(-e\) | \(\pi u_7\) | \(\tau u_6\) | \(-\varepsilon'u_5\) | \(\kappa'u_4\) |
| \(u_4\) | \(u_4\) | \(-\delta'u_5\) | \(\sigma'u_6\) | \(-\pi'u_7\) | \(-e\) | \(\theta u_1\) | \(\omega u_2\) | \(\iota u_3\) |
| \(u_5\) | \(u_5\) | \(\eta u_4\) | \(-\eta' u_7\) | \(-\pi' u_6\) | \(-\theta'u_1\) | \(-e\) | \(\chi u_3\) | \(\zeta u_2\) |
| \(u_6\) | \(u_6\) | \(-\lambda'u_7\) | \(\rho u_4\) | \(\varepsilon u_5\) | \(-\omega'u_2\) | \(-\chi'u_3\) | \(-e\) | \(\nu u_1\) |
| \(u_7\) | \(u_7\) | \(\mu u_6\) | \(\xi u_5\) | \(\kappa u_4\) | \(-\iota'u_3\) | \(-\zeta'u_2\) | \(-\nu'u_1\) | \(-e\) |
Let \(x=x_0e+\sum\limits_{i=1}^{7}x_iu_i\in A\), the endomorphisms \(f: A\longrightarrow A\) and \(g: A\longrightarrow A\) defined by \(f(x)=x_0e+x_1u_1-x_2u_2-x_3u_3+x_4u_4+x_5u_5-x_6u_6-x_7u_7\) and \(g(x)=x_0e+x_1u_1+x_2u_2+x_3u_3-x_4u_4-x_5u_5-x_6u_6-x_7u_7\) are automorphisms of \(A\) which commute. Thus \(Aut(A)\) contains a subgroup isomorphic to the Klein’s group \(\mathbb{Z}_2\times \mathbb{Z}_2\).
The authors did not receive support from any organization for the submitted work.
The authors declare no conflict of interest.