Editing Octonion (section)

===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>