An equivalent definition is to require, for all *x* and *y* in an algebra *A*, that *x*(*xy*) = (*xx*)*y* and (*xy*)*y* = *x*(*yy*). The equivalence of the two definitions is known as Artin's Theorem.

For any two elements *x* and *y* in an alternative algebra another simple identity holds: (*xy*)*x* = *x*(*yx*).

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

Alternativity is a condition in-between associativity and power associativity.