Jacobi identity

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.
The cross product formula_1 and the Lie bracket operation formula_2 both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket.
Definition.
A set A with two binary operations + and "×", with an additive identity 0, satisfies the Jacobi identity if:
The left side is the sum of all even permutations of : the parentheses are left fixed, and letters are interchanged an even number of times.
Commutator bracket form.
The simplest informative example of a Lie algebra is constructed from the (associative) ring of formula_4 matrices, which may be thought of as infinitesimal motions of an "n"-dimensional vector space. The × operation is the commutator, which measures the failure of commutativity in matrix multiplication. Instead of formula_5, the Lie bracket notation is used:
In that notation, the Jacobi identity is:
That is easily checked by computation.
More generally, if A is an associative algebra and "V" is a subspace of A that is closed under the bracket operation: formula_7 belongs to "V" for all formula_8, the Jacobi identity continues to hold on "V". Thus, if a binary operation formula_9 satisfies the Jacobi identity, it may be said that it behaves as if it were given by formula_10 in some associative algebra even if it is not actually defined that way.
Using the antisymmetry property formula_11, the Jacobi identity may be rewritten as a modification of the associative property:
If formula_13 is the action of the infinitesimal motion "X" on "Z", that can be stated as: 
There is also a plethora of graded Jacobi identities involving anticommutators formula_14, such as:
Adjoint form.
Most common examples of the Jacobi identity come from the bracket multiplication formula_16 on Lie algebras and Lie rings. The Jacobi identity is written as:
Because the bracket multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations. Defining the adjoint operator formula_18, the identity becomes:
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:
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 formula_21 map sending each element to its adjoint action is a Lie algebra homomorphism.
Related identities.
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: