Recall that composition algebra $C$ over a field $k$ is specified with a quadratic form $q:C\to k$ . Furthermore, two quadratic forms $q:C\to k$ and $r:D\to k$ are isometric if there exists an invertible linear map $f:C\to D$ such that $r(f(x))=q(x)$ for all $x\in C$ .

A Cayley-Dickson algebra is split if the algebra has non-trivial zero-divisors.

Over the real numbers instead of Witt index, we say the signature of the quadratic form is $(4,4)$ .

This result is often used together with a theorem of Hurwitz which limits the dimensions of composition algebras to dimensions 1,2, 4 or 8. Thus to classify the composition algebras over a given field $k$ of characteristic not 2, it suffices to classify the non-degenerate quadratic forms $q:k^n\to k$ with $n=1,2,4$ or $8$ .

Bibliography

1

Richard D. Schafer, An introduction to nonassociative algebras, Pure and Applied Mathematics, Vol. 22, Academic Press, New York, 1966.