Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2008 / A / 1 / c / Solution

Induct on $i$, proving the claim for all $j$ simultaneously. The case $i=1$ is the definition $[L,L_j]=L_{j+1}$. If it holds for $i$, Jacobi givesBoth uses of the induction hypothesis have first index $i$, so the induction is valid. Thus $\boxed{[L_i,L_j]\subseteq L_{i+j}}$ for all positive indices.