Nilpotent implies solvable

Let $\mathfrak g$ be a finite-dimensional Lie algebra.

Lemma

If $\mathfrak g$ is nilpotent (defined via the lower central series ), then $\mathfrak g$ is solvable (defined via the derived series ).

Context

This inclusion explains why nilpotent Lie algebras sit strictly inside solvable ones; the reverse implication fails in general, and distinguishing solvable from nilpotent is often done by comparing the derived and lower central filtrations.