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 $x,y\in A$.