1. All non-zero nilpotent elements are zero divisors.

2. Unlike the quaternion algebra, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents.

3. The Heisenberg group is a connected nilpotent Lie group of dimension 3, playing a key role in quantum mechanics.

4. If the symmetry algebra is either simple, or nilpotent, then the equations have families of single-valued solutions depending on two arbitrary functions of time.

5. Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.