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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
for all x and y in A.
A composition algebra includes an involution called a conjugation: x ↦ x ∗ . {\displaystyle x\mapsto x^{*}.} The quadratic form N = x x ∗ {\displaystyle N=xx^{*}} is called the norm of the algebra.
A composition algebra is either a division algebra or a split algebra, depending on the existence of a non-zero v in A such that N = 0, called a null vector. When x is not a null vector, the multiplicative inverse of x is x ∗ N {\textstyle {\frac {x^{*}}{N}}}. When there is a non-zero null vector, N is an isotropic quadratic form, and "the algebra splits".