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 which satisfies

for all x and y in A.

A composition algebra includes an involution called a conjugation: x → x*. The quadratic form N(x) = x x*, and is often 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. In case there are no null vectors, the multiplicative inverse of x is x*/N(x), so the algebra is a division algebra. When there is a null vector, N is an isotropic quadratic form, and "the algebra splits".

Every unital composition algebra over a field K can be obtained by repeated application of the Cayley–Dickson construction starting from K (if the characteristic of K is different from 2) or a 2-dimensional composition subalgebra (if char(K) = 2).  The possible dimensions of a composition algebra are 1, 2, 4, and 8.

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