Nilpotent Lie algebra

Summary

A nilpotent Lie algebra is a Lie algebra whose lower central series reaches $0$. Equivalently, all brackets of sufficiently many elements, nested in the form $[x_1, [x_2, [\cdots, x_n]]]$, vanish. Every nilpotent Lie algebra is solvable, but not conversely. By Engel's theorem, a finite-dimensional Lie algebra is nilpotent exactly when each operator $\operatorname{ad} x$ is nilpotent.