Engel subalgebra

Summary

Let $x$ be an element of a finite-dimensional Lie algebra $\mathfrak g$. Its Engel subalgebra is the set of $y \in \mathfrak g$ that are killed by some power of $\operatorname{ad} x$. It is the generalized eigenspace of $\operatorname{ad} x$ for the eigenvalue $0$, and it is a subalgebra containing $x$. Engel subalgebras are the main tool for proving that Cartan subalgebras exist. For a suitably generic $x$, the Engel subalgebra of $x$ is a Cartan subalgebra.