Octonion algebra

Remarks

Every octonion algebra is alternative and not associative. If the characteristic of $K$ is not $2$, it is the Cayley–Dickson double $\mathrm{CD}(Q, \mu)$ of a quaternion algebra $Q$ over $K$ with some $\mu \neq 0$. Two octonion algebras over $K$ are isomorphic if and only if their norms are isometric. An octonion algebra is split if its norm has a nonzero zero, and then it has zero divisors; otherwise it is a division algebra. Example: the octonions $\mathbb{O}$ with $N(x) = x\bar{x}$; this norm is the sum of the squares of the eight coordinates, which is nondegenerate and multiplicative. Non-example: the quaternions $\mathbb{H}$ are a unital composition algebra, but of dimension $4$. Over $\C$ and over finite fields, every octonion algebra is split, so there is only one up to isomorphism.