The mapping given by is a representation of a Lie algebra and is called the adjoint representation of the algebra. (Here, is the Lie algebra of the general linear group over the vector space . It is isomorphic to .)
Within, the composition of two maps is well defined, and the Lie bracket may be shown to be given by the commutator of the two elements,
where denotes composition of linear maps. If is finite-dimensional and a basis for it is chosen, this corresponds to matrix multiplication.
Using this and the definition of the Lie bracket in terms of the mapping ad above, the Jacobi identity
takes the form
where x, y, and z are arbitrary elements of .
This last identity confirms that ad really is a Lie algebra homomorphism, in that the morphism ad commutes with the multiplication operator .
The kernel of is, by definition, the center of .
Read more about this topic: Adjoint Endomorphism