This lecture is part of an online graduate course on Lie groups.
We state Engel's theorem about nilpotent Lie algebras and sketch a proof of it. We give an example of a nilpotent Lie group that is not a matrix group.
For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53RWBkiHKoOsTw-dGHAoJ-h
