Killing Form

The Killing form is an inner product on a finite dimensional Lie algebra defined by

(1)

in the adjoint representation, where is the adjoint representation of . (1) is adjoint-invariant in the sense that

(2)

When is a semisimple Lie algebra, the Killing form is nondegenerate.

For example, the special linear Lie algebra has three basis vectors , where :

			(3)
			(4)
			(5)

The other brackets are given by and . In the adjoint representation, with the ordered basis , these elements are represented by

			(6)
			(7)
			(8)

and so where

(9)