[Hurwitz's theorem](https://en.wikipedia.org/wiki/Hurwitz%27s_theorem_(composition_algebras)) states that a real possibly-non-associative algebra (meaning just a real vector space $V$ equipped with a map $m: V \otimes V \to V$) along with a *positive definite* quadratic form $|\cdot|^2 : V \to \mathbb R$ which is multiplicative (in the sense that $|m(v,w)|^2 = |v|^2 |w|^2$) is necessarily one of the algebras $\mathbb R$, $\mathbb C$, $\mathbb H$, or $\mathbb O$.


What happens when positive definiteness is dropped? (To make the question nontrival, let's demand that the associated symmetric pairing on $V$ is nondegenerate.)



Normed algebras of indefinite signature?
I believe the answer to my question is that these are called [composition algebras](https://en.wikipedia.org/wiki/Composition_algebra) and are essentially fully understood. I was hoping the answer would be more interesting.


