Alternative algebra

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In abstract algebra, an algebra is called alternative if (xx)y = x(xy) and y(xx) = (yx)x for all x and y in the algebra, that is, if the multiplication is alternative.

Equivalently, an algebra is alternative if and only if the subalgebra generated by any two of its elements is associative. The equivalence of the two definitions is known as Artin's theorem, after Emil Artin.

For any two elements x and y in an alternative algebra another simple identity holds: (xy)x = x(yx). This is called the flexible law.

Every associative algebra is obviously alternative, but so too are some non-associative algebras such as the octonions. The sedenions are not alternative.

Alternativity in algebras is a condition weaker than associativity but stronger than power associativity.