Engel's theorem states that every finite-dimensional Engel algebra is nilpotent.

(Engel's theorem)

In fact, if has only nilpotent elements, then it is nilpotent (Engel's theorem), but then its Killing form is identically zero, contradicting semisimplicity.

The key lemma in the proof of Engel's theorem is the following fact about Lie algebras of linear operators on finite dimensional vector spaces which is useful in its own right:

In representation theory, a branch of mathematics, Engel's theorem is one of the basic theorems in the theory of Lie algebras; it asserts that for a Lie algebra two concepts of nilpotency are identical.

A central series is analogous in Lie theory to a flag that is strictly preserved by the adjoint action (more prosaically, a basis in which each element is represented by a strictly upper triangular matrix); compare Engel's theorem.

In fact, by Engel's theorem, any finite-dimensional nilpotent Lie algebra is conjugate to a subalgebra of the strictly upper triangular matrices, that is to say, a finite-dimensional nilpotent Lie algebra is simultaneously strictly upper triangularizable.

By Engel's theorem, if is a semidirect product, with abelian and nilpotent, acting on a finite dimensional vector space W with operators in diagonalizable and operators in nilpotent, there is a vector w that is an eigenvector for and is annihilated by .