Lower central series of a group (reviewed)

If $G$ is a group, for subsets $A$ and $B$ let $[A,B]$ be the subgroup generated by commutators of $A$ and $B$, i.e., \[ [A,B] = \langle a^{-1}b^{-1}ab\mid a\in A \text{ and } b\in B\rangle.\] Further, let $G^0=G$ and for $i>0$, $G^i = [G,G^{i-1}]$. Then, the lower central series of $G$ is a normal series of subgroups \[ G \unrhd G^1 \unrhd G^2 \unrhd \cdots .\] Then, $G$ is nilpotent if and only if $G^k =\langle e\rangle$ for some $k>0$.