Let $G$ be a group generated by $S=\{x_1,\cdots,x_n\}$ and let its lower central series be defined as $\Gamma_1=G$, $\Gamma_m=[\Gamma_{m-1},G]$ for $m\geq 2$.
By definition $\Gamma_m$ is generated by all iterated commutators of the form $[[g_1,g_2],\cdots,g_m]$, where $g_1,\cdots,g_m\in G$.
Is it true that $\Gamma_m$ is generated by all iterated commutators of the form $[[x_{i_1},x_{i_2}],\cdots,x_{i_m}]$, where $x_{i_1},\cdots,x_{i_m}\in S$?
This is not true, even for $\Gamma_2$. Let $G$ be the free group on $S$; it is known that $[G,G]$ is not finitely generated (provided that $\vert S\vert\ge 2$), hence it cannot be generated by all commutators of the form $[x_{i_1}, x_{i_2}]$ since there are only finitely many such.