Anticommutative property

In mathematics, anticommutativity is a specific property of some non-commutative operations. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation, such as addition.

Subtraction is an anticommutative operation because −(ab) = ba. For example, 2 − 10 = −(10 − 2) = −8.

A prominent example of an anticommutative operation is the Lie bracket.

and the proof in the multilinear case is the same but in only two of the inputs.