Here by $b(n,F)$ we mean the Lie algebra of all $n \times n$ upper triangular matrices. So, the first step of the question asks to prove that $L^m$ has a basis consisting of all matrix units $e_{ij}$ with $j - i > m$.
So I thought I would try to compute $L^m$ for different $m$. Now, $e_{ij}$ where $i < j$ form the basis for this Lie algebra. $[e_{ij}, e_{kl}]$ $=$ $\delta_{jk}e_{il}$ - $\delta_{il}e_{kj}$. So we have two cases, $j = k$ or $k = l$, but not both. These two cases lead to $L^1$ being the span of $e_{il}$ and $e_{kj}$. Now I don't know how to check if the claim in the question is true for $L^1$. Can anyone help me out?
