Jacobson’s theorem on composition algebras

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 aninvertible linear map $f:C\to D$ such that $r(f(x))=q(x)$ for all $x\in C$.

Theorem 1 (Jacobson).

[1, Theorem 3.23]Two unital Cayley-Dickson algebras $C$ and $D$ over a field $k$ of characteristic not $\mathrm{2}$are isomorphic if, and only if, their quadratic forms are isometric.

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

Corollary 2.

[1, Corollary 3.24]Upto isomorphism there is only one split Cayley-Dickson algebra and the quadratic formhas Witt index 4.

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 dimensionsof composition algebras to dimensions 1,2, 4 or 8. Thus to classify the composition algebrasover a given field $k$ of characteristic not 2, it suffices to classify the non-degeneratequadratic forms $q:k^n\to k$ with $n=1,2,4$ or $8$.