Anticommutative algebra

Remarks

Expanding $(x + y)(x + y) = 0$ gives $xy = -yx$ for all $x, y \in A$. Conversely, $xy = -yx$ for all $x, y$ gives $2xx = 0$, so the two conditions agree when $2$ is invertible in $R$. Example: $\R^3$ with the cross product, since $v \times v = 0$ for every vector $v$. Non-example: the field $\mathbb{F}_2$ as an algebra over itself satisfies $xy = -yx$, because it is commutative and $-1 = 1$, but $1 \cdot 1 = 1 \neq 0$. Some authors call an algebra with $xy = -yx$ anticommutative and one with $xx = 0$ alternating.