= 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 of a Lie algebra>[ideal]. Inductively, if $L_n$ is an <ideal of a Lie algebra>[ideal], the <Jacobi identity> gives
$$
[z,[x,y]]=[[z,x],y]+[x,[z,y]]\in[L,L_n]\quad(z,x\in L,\ y\in L_n),
$$
so $L_{n+1}$ is an <ideal of a Lie algebra>[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 $L_c\ne0$ for nonzero $L$.
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