Lie Algebra Commutator Series

The Lie algebra commutator series of a Lie algebra , sometimes called the derived series, is the sequence of subalgebras recursively defined by

(1)

with . The sequence of subspaces is always decreasing with respect to inclusion or dimension, and becomes stable when is finite dimensional. The notation means the linear span of elements of the form , where and .

When the commutator series ends in the zero subspace, the Lie algebra is called solvable. For example, consider the Lie algebra of strictly upper triangular matrices, then

			(2)
			(3)
			(4)

and . By induction, where is the term in the Lie algebra lower central series.

In contrast to the solvable Lie algebras, the semisimple Lie algebras over a field of field characteristic zero have a constant commutator series. An intermediate example for is

(5)

where is semisimple. The matrix trace identity

(6)

shows that every commutator has matrix trace zero. Here, is a general linear Lie algebra and is the special linear Lie algebra.