Anticommutative algebra

Summary

An anticommutative algebra is a nonassociative algebra in which the square of every element is zero. It follows that $xy = -yx$ for all $x$ and $y$: swapping the factors changes the sign. Lie algebras and $\R^3$ with the cross product are the basic examples. Anticommutativity is far from associativity, so such algebras are studied with their own identities, such as the Jacobi identity.