${\frak h}_n(\mathbb{C})$ is a solvable lie algebra

Question

Can anyone give me a suggestion to solve this problem?

Show that $${\frak h}_n(\mathbb{C}) = \lbrace A\in\operatorname{Mat}_n(\mathbb{C}) : A_{i,j} = 0\text{ if } i\geqslant j \rbrace.$$

is solvable Lie algebra.

Answer 1

These are lower triangular matrices. Your $\mathfrak{h}_n$ has a basis consisting of $E_{i,j}$ where $i-j\ge1$. Denote the space of matrices spanned by the $E_{i,j}$ with $i-j\ge k$ as $L_k$. Show that $[L_k,L_k] \subseteq L_{2k}$.