Alternative algebra

Summary

An alternative algebra is a nonassociative algebra in which $x(xy) = (xx)y$ and $(yx)x = y(xx)$ for all $x, y$. Equivalently, the associator is alternating: it vanishes when two of its arguments are equal, and so it changes sign when two arguments are swapped. The octonions are the main example that is not associative. By Artin's theorem, any two elements generate an associative subalgebra. The Moufang identities also hold in every alternative algebra.