Editing Jacobi identity (section)

==Adjoint form==
Most common examples of the Jacobi identity come from the bracket multiplication <math>[x,y]</math> on [[Lie algebra]]s and [[Lie ring]]s. The Jacobi identity is written as:

: <math>[x,[y,z]] + [z,[x,y]] + [y,[z,x]] = 0.</math>

Because the bracket multiplication is [[anticommutativity|antisymmetric]], the Jacobi identity admits two equivalent reformulations. Defining the [[adjoint representation of a Lie algebra|adjoint operator]] <math>\operatorname{ad}_x: y \mapsto [x,y]</math>, the identity becomes:
:<math>\operatorname{ad}_x[y,z]=[\operatorname{ad}_xy,z]+[y,\operatorname{ad}_xz].</math>

Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a [[derivation (abstract algebra)|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:
:<math>\operatorname{ad}_{[x,y]}=[\operatorname{ad}_x,\operatorname{ad}_y].</math>

There, the bracket on the left side is the operation of the original algebra, the bracket on the right is the commutator of the composition of operators, and the identity states that the <math>\mathrm{ad}</math> map sending each element to its adjoint action is a [[Lie algebra homomorphism]].