Composition algebra

Definition
Suppose $$K$$ is a field. A composition algebra or normed algebra over $$K$$ is a unital non-associative (in the sense of not necessarily associative) algebra $$A$$ over $$K$$ equipped with a function $$N:A \to K$$ such that:


 * $$N$$ is a nondegenerate quadratic form on $$A$$, viewed as a vector space over $$K$$. In other words, $$N(\alpha x) = \alpha^2 N(x)$$ for all $$\alpha \in K, x \in A$$, and the function $$(x,y) \mapsto N(x + y) - N(x) - N(y)$$ is a nondegenerate bilinear form on $$A$$.
 * $$N$$ is a homomorphism of unital magmas from $$A$$ to $$K$$, i.e., $$N(1) = 1$$ and $$N(xy) = N(x)N(y)$$ for all $$x,y \in A$$.