Editing Octonion (section)

==Properties==
Octonionic multiplication is neither [[commutative]]:

:{{math|''e{{sub|i}} e{{sub|j}}'' {{=}} −''e{{sub|j}} e{{sub|i}}'' ≠ ''e{{sub|j}} e{{sub|i}}''}} if {{mvar|i}}, {{mvar|j}} are distinct and non-zero,

nor [[associative]]:

:{{math|(''e{{sub|i}} e{{sub|j}}'') ''e{{sub|k}}'' {{=}} −''e{{sub|i}}'' (''e{{sub|j}} e{{sub|k}}'') ≠ ''e{{sub|i}}''(''e{{sub|j}} e{{sub|k}}'')}} if {{mvar|i}}, {{mvar|j}}, {{mvar|k}} are distinct, non-zero and {{math|''e{{sub|i}} e{{sub|j}}'' ≠ ±''e{{sub|k}}''}}.

The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of <math>\ \mathbb{O}\ </math> is [[isomorphic]] to [[real numbers|{{math|ℝ}}]], [[complex numbers|{{math|ℂ}}]], or [[quaternions|{{math|ℍ}}]], all of which are associative. Because of their non-associativity, octonions cannot be represented by a subalgebra of a [[matrix ring]] over {{math|ℝ}}, unlike the real numbers, complex numbers, and quaternions.

The octonions do retain one important property shared by {{math|ℝ}}, {{math|ℂ}}, and {{math|ℍ}}: the norm on <math>\ \mathbb{O}\ </math> satisfies

:<math> \| x y \| = \| x \|\ \| y \| ~.</math>

This equation means that the octonions form a [[composition algebra]]. The higher-dimensional algebras defined by the Cayley–Dickson construction (starting with the [[sedenion]]s) all fail to satisfy this property. They all have [[zero divisor]]s.

Wider number systems exist which have a multiplicative modulus (for example, 16&nbsp;dimensional conic sedenions). Their modulus is defined differently from their norm, and they also contain zero divisors.

As shown by [[Adolf Hurwitz|Hurwitz]], {{math|ℝ}}, {{math|ℂ}}, or {{math|ℍ}}, and <math>\ \mathbb{O}\ </math> are the only normed division algebras over the real numbers. These four algebras also form the only alternative, finite-dimensional [[division algebra]]s over the real numbers ([[up to]] an isomorphism).

Not being associative, the nonzero elements of <math>\ \mathbb{O}\ </math> do not form a [[Group (mathematics)|group]]. They do, however, form a [[loop (algebra)|loop]], specifically a [[Moufang loop]].

===Commutator and cross product===
The [[commutator]] of two octonions {{mvar|x}} and {{mvar|y}} is given by

:<math>[x, y] = xy - yx ~.</math>

This is antisymmetric and imaginary. If it is considered only as a product on the imaginary subspace <math>\ \operatorname\mathcal{I_m}\bigl[\ \mathbb{O}\ \bigr]\ </math> it defines a product on that space, the [[seven-dimensional cross product]], given by

:<math>x \times y = \tfrac{\ 1\ }{ 2 }\ (xy - yx) ~.</math>

Like the [[cross product]] in three dimensions this is a vector orthogonal to {{mvar|x}} and {{mvar|y}} with magnitude

:<math>\|x \times y\| = \|x\|\ \|y\|\ \sin \theta ~.</math>

But like the octonion product it is not uniquely defined. Instead there are many different cross products, each one dependent on the choice of octonion product.<ref>{{harvp|Baez|2002|pp=37–38}}</ref>

===Automorphisms===
An [[automorphism]], {{mvar|A}}, of the octonions is an invertible [[linear transformation]] of <math>\ \mathbb{O}\ </math> which satisfies

:<math>A(xy) = A(x)\ A(y) ~.</math>

The set of all automorphisms of <math>\ \mathbb{O}\ </math> forms a group called {{nobr|{{math|[[G2 (mathematics)|''G''{{sub|2}}]]}} .}}<ref>{{harv|Conway|Smith|2003|loc=ch&nbsp;8.6}}</ref> The group {{math|''G''{{sub|2}} }} is a [[simply connected]], [[Compact group|compact]], real [[Lie group]] of dimension&nbsp;14. This group is the smallest of the exceptional Lie groups and is isomorphic to the [[subgroup]] of {{math|Spin(7)}} that preserves any chosen particular vector in its 8&nbsp;dimensional real spinor representation. The group {{math|Spin(7)}} is in turn a subgroup of the group of isotopies described below.

''See also'': {{math|[[PSL(2,7)]]}} – the [[automorphism group]] of the Fano plane.

===Isotopies===
An [[isotopy of an algebra]] is a triple of [[bijection|bijective]] linear maps {{mvar|a}}, {{mvar|b}}, {{mvar|c}} such that if {{math|''xy'' {{=}} ''z''}} then {{math|''a''(''x'')''b''(''y'') {{=}} ''c''(''z'')}}. For {{math|''a'' {{=}} ''b'' {{=}} ''c''}} this is the same as an automorphism. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup.

The isotopy group of the octonions is the group {{math|Spin<sub>8</sub>(ℝ)}}, with {{mvar|a}}, {{mvar|b}}, {{mvar|c}} acting as the three 8&nbsp;dimensional representations.<ref>{{harv|Conway|Smith|2003|loc=ch&nbsp;8}}</ref> The subgroup of elements where {{mvar|c}} fixes the identity is the subgroup {{math|Spin<sub>7</sub>(ℝ)}}, and the subgroup where {{mvar|a}}, {{mvar|b}}, {{mvar|c}} all fix the identity is the automorphism group {{nobr|{{math|''G''{{sub|2}} }} .}}

===Matrix representation===
Just as quaternions can be [[Quaternion#Matrix_representations|represented as matrices]], octonions can be represented as tables of quaternions. Specifically, because any octonion can be defined a pair of quaternions, we represent the octonion <math> ( q_0, q_1 )</math> as:
<math display=block>\begin{bmatrix}
         q_0 & q_1 \\
        -q_1^* & q_0^*
\end{bmatrix}</math>

Using a slightly modified (non-associative) quaternionic matrix multiplication:
<math display=block>\begin{bmatrix}
         \alpha_0 & \alpha_1 \\
        \alpha_2 & \alpha_3
\end{bmatrix}\circ\begin{bmatrix}
         \beta_0 & \beta_1 \\
        \beta_2 & \beta_3
\end{bmatrix}=\begin{bmatrix}
         \alpha_0\beta_0+\beta_2\alpha_1 & \beta_1\alpha_0+\alpha_1\beta_3\\
        \beta_0\alpha_2+\alpha_3\beta_2 & \alpha_2\beta_1+\alpha_3\beta_3
\end{bmatrix}</math>
we can translate octonion addition and multiplication to the respective operations on quaternionic matrices.<ref name="Ensembles"></ref>