nLab
lower central series

Given a group $G$, its lower central series is the inductively defined descending sequence

$$G=G_0\supset G_1\supset G_2\supset \dots$$G = G_0 \supset G_1\supset G_2\supset \ldots

in which $G_k=[G,G_{k-1}]$ is the subgroup generated by all commutators ${\mathrm{ghg}}^{-1}{h}^{-1}$ where $g\in G$ and $h\in G_{k-1}$.

Similarly, given a Lie algebra $L$, its lower central series is the inductively defined descending sequence of Lie subalgebras $L=L_0\supset L_1\supset L_2\supset \dots$ in which $L_k=[L,L_{k-1}]$ is the Lie subalgebra generated bt all commutators $[l,h]$ where $l\in L$ and $h\in L_{k-1}$.