Solution
= 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 gives
$$
[L_{i+1},L_j]=[[L,L_i],L_j]\subseteq[L,[L_i,L_j]]+[L_i,[L,L_j]]\subseteq[L,L_{i+j}]+[L_i,L_{j+1}]\subseteq L_{i+j+1}.
$$
Both 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.