nLab
alternative algebra

A magma $A$ is (left and right) alternative if its product satisfies

• left alternativity $x(xy)=(xx)y$ for all $x,y\in A$
• right alternativity $(xy)y=x(yy)$ for all $x,y\in A$

A nonassociative algebra $(A,\cdot ,+)$ is alternative if its underlying magma $(A,\cdot )$ alternative.

In particular, every associative algebra is alternative. The properly nonassociative algebra of octonions is also alternative. The reason is that it is obtained by a doubling procedure from the algebra of quaternions which is associative; and the double of any associative $R$-algebra with involution is alternative.