Composition algebra

Summary

A composition algebra is an algebra over a field with a nondegenerate quadratic form $N$ that is multiplicative: $N(xy) = N(x)N(y)$. The model is $\C$ with $N(z) = \abs{z}^2$, where multiplicativity is the rule $\abs{zw}^2 = \abs{z}^2\abs{w}^2$. By Hurwitz's theorem, a composition algebra with a unit has dimension $1$, $2$, $4$ or $8$. Over a field of characteristic not $2$, every unital one is obtained from the field by repeated Cayley–Dickson construction. The quaternions, the octonions and their split forms are composition algebras, and so is the non-unital Okubo algebra.