In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
 x(xy) = (xx)y
 (yx)x = y(xx)
for all x and y in the algebra. Every associative algebra is obviously alternative, but so too are some strictly nonassociative algebras such as the octonions. The sedenions, on the other hand, are not alternative.
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The associator
Alternative algebras are so named because they are precisely 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 it arguments are equal. The left and right alternative identities for an algebra are equivalent to
Both of these identities together imply that the associator is totally skewsymmetric. That is,
for any permutation σ. It follows that
for all x and y. This is equivalent to the socalled flexible identity
The associator is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
 left alternative identity: x(xy) = (xx)y
 right alternative identity: (yx)x = y(xx)
 flexible identity: (xy)x = x(yx).
is alternative and therefore satisfies all three identities.
An alternating associator is always totally skewsymmetric. The converse holds so long as the characteristic of the base field is not 2.
Properties
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative. Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written without parenthesis unambiguously in an alternative algebra. A generalization of Artin's theorem states that whenever three elements x,y,z in an alternative algebra associate (i.e. [x,y,z] = 0) the subalgebra generated by those elements is associative.
A corollary of Artin's theorem is that alternative algebras are powerassociative, that is, the subalgebra generated by a single element is associative. The converse need not hold: the sedenions are powerassociative but not alternative.
The Moufang identities
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