In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have

for all x and y in the algebra.

Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions.

Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by

By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to

Both of these identities together imply that:

for all ${\displaystyle x}$ and ${\displaystyle y}$. This is equivalent to the flexible identity

The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of: