Let $L$ be a Lie algebra over a field $k$ and $m, n$ be positive integers. I am trying to prove that $$[L^m, L^n] \subseteq L^{m+n}. $$
I tried to use induction. The base step is immediate by definition. Assume that $[L^k, L^n] \subseteq L^{k+n},$ for all $n$.
Then we take $[L^{k+1}, L^n] = [[L, L^k] , L^n] $ and we should prove that it is contained in $L^{n+k+1}. $
The hint I was given is that the following relation holds: $$[[L, L^k], L^n] \subseteq [L, [L, ^k, L^n]] +[L^k, [L, L^n]]. $$
And indeed, if we prove this, the conclusion is easy.
But I am struggling and still can't prove the hint. Can anyone write down a complete proof of why this equation holds?
Note that we define $[I, J] :=\mathrm{span} \{[x, y] :x\in I, y\in J\}$ and $L^{n+1}=[L,L^n]$.
