A binary operation “$\star$” is said to be anticommutative if it satisfies the identity

where the minus denotes the opposite element in the algebra in question.  This implies that  $x\!\star\!x\;=\;-(x\!\star\!x)$,  i.e. $x\!\star\!x$ must be the neutral element of the addition of the algebra:

Using the distributivity of “$\star$” over “$+$” we see that the indentity (2) also implies (1):

A well known example of anticommutative operations is the vector product in the algebra  $(\mathbb{R}^{3},\,+,\,\times)$,  satisfying

Also we know that the subtraction of numbers obeys similar identities

An important anticommutative operation is the Lie bracket.