Representations of Nilpotent Lie Subalgebras

Question

Does every nilpotent lie subalgebra of gl(V) is representable by triangular matrices with 0 on the diagonal?

Answer 1

This is not true for rather simple reasons: Any one-dimensional subspace in $\mathfrak{gl}(V)$ is an Abelian (and hence nilpotent) Lie subalgebra. This can be represented by upper triangular matrices with $0$ on the diagonal if and only if it is spanned by a nilpotent map.

Engel's theorem as mentioned in the comment by @peter_a_g is not a theorem about nipotent Lie subalgebras in $\mathfrak{gl}(V)$ but about Lie subalgebras consisting of nilpotent endomorphisms of $V$. (It then implies that such a subalgebra is nilpotent, but the converse is not true.)