Let $M=L_r$ and write $M_k$ for its own lower central series. Part (c) proves inductively that $M_k\subseteq L_{rk}$: it holds at $k=1$, and $[M,M_k]\subseteq[L_r,L_{rk}]\subseteq L_{r(k+1)}$. If $L$ has class less than $rs$, then $L_{rs}=0$, so $M_s=0$. Consequently $\boxed{\operatorname{class}(L_r)<s}$, with the zero algebra understood to have class zero.

Solution

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

 Solution (source code)