Jacobi identity

Property of some binary operations, such as the cross product and any ring's commutator

In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician Carl Gustav Jakob Jacobi.

Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a derivation. That form of the Jacobi identity is also used to define the notion of Leibniz algebra.

Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation:

The Hall–Witt identity is the analogous identity for the commutator operation in a group.

The following identitity follows from anticommutativity and Jacobi identity and holds in arbitrary Lie algebra:[2]