Alternative algebra

Remarks

In terms of the associator, the laws say $(x, x, y) = 0$ and $(y, x, x) = 0$. Replacing $x$ by $x + z$ gives $(x, z, y) = -(z, x, y)$ and $(y, x, z) = -(y, z, x)$. So the associator is alternating: it changes sign under any swap of two arguments and vanishes when two arguments are equal. In particular $(x, y, x) = 0$, so an alternative algebra is flexible. Example: every associative algebra is alternative, and so are the octonions. Non-example: $\R^3$ with the cross product is not alternative, since $(e_1 \times e_1) \times e_2 = 0$ but $e_1 \times (e_1 \times e_2) = -e_2$.