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

Define $L_1=L$ and $L_{n+1}=[L,L_n]$, where brackets of subspaces mean their linear span. Each term is an ideal. Inductively, if $L_n$ is an ideal, the Jacobi identity givesso $L_{n+1}$ is an ideal as well. Also $L_{n+1}\subseteq L_n$. Nilpotence means that this descending series eventually reaches zero; class $c$ means $L_{c+1}=0$ with $L_c\ne 0$ for nonzero $L$.