I want some example of nilpotent lie algebras, here I also want to see how the set of all $n\times n$ matrices $(a_{ij})$ where $a_{ij} = 0\ \forall\ i\ge j$ forms a nilpotent lie algebra under the lie multiplication $[AB]=AB-BA$.
I can visualize that a matrix of that form when raised to power $n$ gives $0$ matrix. are they lie group? I guess not because as they are nilpotent matrix they are closed subset of $M_n(R)$ right? hence they are not manifold hence not a lie group?