# Determine the derived series of $\mathfrak{b}_n (\mathbb{C})$. [duplicate]

Problem: Determine the derived series of $$\mathfrak{b}_n (\mathbb{C})$$, in which $$\mathfrak{b}_n (\mathbb{C})$$ is the space of all upper triangular matrices.

We knew that the derived series of a Lie algebra $$L$$ is $$L^{(0)}=L, L^{(1)}=[LL],L^{(2)}=[L^{(1)}L^{(1)}], \dots, L^{(i)}=[L^{(i-1)}L^{(i-1)}]$$.

How do I finding the derived series of $$\mathfrak{b}_n (\mathbb{C})$$?

First edited: $$L = \mathfrak{b}_n (\mathbb{C})$$, so for $$n=2$$, the basis of $$\mathfrak{b}_2 (\mathbb{C})$$ is $$e_1=\begin{pmatrix}1 & 0\\ 0 & 0 \end{pmatrix}, e_2 =\begin{pmatrix}0 & 1\\ 0 & 0 \end{pmatrix}, e_3=\begin{pmatrix}0 & 0\\ 0 & 1 \end{pmatrix}$$

$$[e_1 e_2] = e_1, [e_1 e_3]=0, [e_2 e_3]=e_3$$

$$L^{(1)}=[LL] = \langle e_1,e_3 \rangle$$

$$L^{(2)}=[L^{(1)} L^{(1)}] = 0$$

How to continue the computation?

## marked as duplicate by Dietrich Burde, Torsten Schoeneberg, Robert Soupe, Shailesh, Lord Shark the UnknownApr 29 at 4:37

By definition $$\mathfrak b_n(\mathbb C)$$ is the set of all $$n\times n$$ complex matrices $$(a_{jk})_{1\leqslant j,k\leqslant n}$$ such that $$a_{jk}=0$$ whenever $$j>k$$. It turns out that, for each $$l\in\{1,2,\ldots,n\}$$,$$\mathfrak b_n^{(l)}(\mathbb C)=\left\{\text{matrices }(a_{jk})_{1\leqslant j,k\leqslant n}\,\middle|\,j+l>k\implies a_{jk}=0\right\}.$$Prove this by induction on $$l$$ and you're done.