Octonion algebra

Summary

An octonion algebra over a field $K$ is a unital composition algebra of dimension $8$ over $K$. It is alternative and not associative, and it is the Cayley–Dickson double of a quaternion algebra. The octonions and the split-octonions are the two octonion algebras over $\R$. Octonion algebras are tied to the exceptional Lie group $G_2$, which is their automorphism group, and to the exceptional Jordan algebras.