Now that I’ve discussed some of the basic definitions in the theory of Lie algebras, it’s time to look at specific subclasses: nilpotent, solvable, and eventually semisimple Lie algebras. Today, I want to focus on nilpotence and its applications.

Engel’s Theorem

To start with, choose a Lie algebra {L \subset \mathfrak{gl} (V)} for some finite-dimensional {k}-vector space {V}; recall that {\mathfrak{gl} (V)} is the Lie algebra of linear transformations {V \rightarrow V} with the bracket {[A,B] := AB - BA}. The previous definition was in terms of matrices, but here it is more natural to think in terms of linear transformations without initially fixing a basis.

Engel’s theorem is somewhat similar in its statement to the fact that commuting diagonalizable operators can be simultaneously diagonalized.