Nilpotent Lie group

Summary

A nilpotent Lie group is a connected Lie group that is a nilpotent group. So its lower central series reaches the trivial group after finitely many steps. Equivalently, its Lie algebra is a nilpotent Lie algebra. Every nilpotent Lie group is a solvable Lie group, but not conversely. For a connected and simply connected nilpotent Lie group, the exponential map from the Lie algebra is a diffeomorphism. So such a group can be studied almost entirely through its Lie algebra.