Examples
The Jacobi identity is satisfied by the multiplication (bracket) operation on Lie algebras and Lie rings and these provide the majority of examples of operations satisfying the Jacobi identity in common use. Because of this the Jacobi identity is often expressed using Lie bracket notation:
If the multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations. Defining the adjoint map
after a rearrangement, the identity becomes
Thus, the Jacobi identity for Lie algebras simply becomes the assertion that the action of any element on the algebra is a derivation. This 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:
This identity implies that the map sending each element to its adjoint action is a Lie algebra homomorphism of the original algebra into the Lie algebra of its derivations.
A similar identity, called the Hall–Witt identity, exists for the commutators in groups.
In analytical mechanics, Jacobi identity is satisfied by Poisson brackets. In the Copenhagen interpretation of quantum mechanics, it is satisfied by operator commutators on Hilbert space and, equivalently, in the phase space formulation of quantum mechanics by the Moyal bracket.
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