Engel's theorem

This fact is related to: Lie algebras
View other facts related to Lie algebrasView terms related to Lie algebras |

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.