Composition algebra

In mathematics, a composition algebra A over a field K is a not necessarily associative algebra over K together with a nondegenerate quadratic form N that satisfies

${\displaystyle N(xy)=N(x)N(y)}$

for all x and y in A.

A composition algebra includes an involution called a conjugation: ${\displaystyle x\mapsto x^{*}.}$ The quadratic form ${\displaystyle N(x)=xx^{*}}$ is called the norm of the algebra.

A composition algebra (A, ∗, N) is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N(v) = 0, called a null vector.^[1] When x is not a null vector, the multiplicative inverse of x is ${\textstyle {\frac {x^{*}}{N(x)}}}$. When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits".