Engel's theorem

Statement
Here are two versions. The first is the basic statement, and the second follows:


 * 1) Let $$V$$ be a finite-dimensional vector space (over any field), and $$L$$ a Lie subalgebra of $$gl(V)$$ such that every element of $$L$$ induces a nilpotent linear transformation on $$V$$. Then, there is a basis for $$V$$ in which all elements of $$L$$ are strictly upper triangular matrices.
 * 2) If $$L$$ is a finite-dimensional Lie algebra over a field such that for any $$x \in L$$, $$ad \, x$$ is nilpotent, then $$L$$ is a nilpotent Lie algebra. In other words, for finite-dimensional Lie algebras over a field, being an Engel Lie ring is equivalent to being a nilpotent Lie ring.

Related facts

 * Lie's theorem: Analogous statement for solvable Lie algebras.
 * Kolchin's theorem: Analogous statement for unipotent transformations in algebraic groups
 * Lie-Kolchin theorem: Analogous statement for solvable groups.
 * Kostrikin's theorem
 * Zelmanov's theorem: Analogous statement for arbitrary Lie rings.

Facts
Engel's theorem does not say that any nilpotent Lie algebra inside $$gl(V)$$ can be given a basis where it is upper triangular. For instance, a one-dimensional Lie algebra is Abelian and hence nilpotent, yet one may not be able to change basis to make it strictly upper triangular.