# Engel's theorem

From Groupprops

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:

- Let be a finite-dimensional vector space (over any field), and a Lie subalgebra of such that every element of induces a nilpotent linear transformation on . Then, there is a basis for in which all elements of are strictly upper triangular matrices.
- If is a finite-dimensional Lie algebra over a field such that for any , is nilpotent, then 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 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.