Composition algebra

Remarks

$N$ is called the norm of $A$. A composition algebra with a unit is called a Hurwitz algebra or a unital composition algebra. In a unital composition algebra each $x$ satisfies $x^2 - b(x, 1)x + N(x)1 = 0$. Example: $\C$ over $\R$ with $N(z) = \abs{z}^2$; the polar form is $2\operatorname{Re}(z\bar{w})$, which is nondegenerate, and $\abs{zw}^2 = \abs{z}^2\abs{w}^2$. Non-example: $\R^3$ with the cross product and $N(v) = v \cdot v$ fails clause 2, since $N(e_1 \times e_1) = 0$ while $N(e_1)N(e_1) = 1$. Two unital composition algebras are isomorphic if and only if their norms are isometric.