On nilpotent Lie algebras arising from parabolic subalgebras and quivers --- three-step cases

Nilpotent Lie groups are important subjects for studying spacial left-invariant geometric structures. However, nilpotent Lie groups are impossible to classify, and even the two-step case has not been classified. There are several classes of examples of nilpotent Lie algebras that arise with higher nilpotency steps. In this paper, we focus on two of these methods. The first one is the nilpotent parts of parabolic subalgebras of semisimple Lie algebras. The second one is the nilpotent Lie algebras obtained by quivers. We list and compare nilpotent Lie algebras obtained by these two methods that are three-step and of dimension at most 10.