Jacobi identity /

In mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity. A binary operation * on a set S possessing a commutative binary operation + with additive identity 0 satisfies the Jacobi identity if 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 commutator. The Jacobi Identity can then be translated into words: "the infinitesimal motion of B followed by the infinitesimal motion of A (), minus the infinitesimal motion of A followed by the infinitesimal motion of B (), is the infinitesimal motion of [A,B] (), when acting on any arbitrary infinitesimal motion C (thus, these are equal)". 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