Jacobi identity
In mathematics the Jacobi identity is a property a binary operation can have that determines how the order of evaluation behaves for the given operation. In contrast to plain associative operations, the order of evaluation is significant for operations satisfying the Jacobi identity. It is named after the German mathematician Carl Gustav Jakob Jacobi. The cross product and the bracket operation of a Lie algebra both satisfy the Jacobi identity.
A binary operation × on a set S possessing a binary operation + with an additive identity denoted by 0 satisfies the Jacobi identity if
That is, the sum of all even permutations of (a,(b,c)) must be zero (where the permutation is performed by leaving the parentheses fixed and interchanging letters an even number of times).
In a Lie algebra, the objects that obey the Jacobi identity are infinitesimal motions. When acting on an operator with an infinitesimal motion, the change in the operator is the ring commutator.
The Jacobi Identity is manifestly cyclic,
![{\displaystyle [A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/046dbc0d4387302c8d321afb443e35bc1890fcac)
...
Wikipedia