lower central series and basic generators

Question

After reading the following question:

Third quotient of the lower central series

I have a similar question.

Let us denote $F_n$ the free group with $n$ generators, $x_1,...,x_n$ and let $[a,b] = a^{-1}b^{-1}ab$, and $G_i$ be the $i^{th}$ group in the lower central series. It is known that $G_3/G_4$ is generated by (the basic commutators) $[[x_i,x_j],x_k]$ where $i>j$ and $j \leq k$. Now, let $i,j,l,k$ be indices. How do we express $[[x_i,x_jx_l],x_k]$ in $G_3/G_4$ using only elements of the form $[[a,b],c]$ when $a,b,c \in \{x_i,x_j,x_l,x_k\}$?

Thank you, David

Answer 1

$[[x_i,x_jx_l],x_k] = [[x_i,x_l][x_i,x_j]^{x_l},x_k] =$ $[[x_i,x_l][x_l,[x_i,x_j]^{-1}][x_i,x_j],x_k] \equiv [[x_i,x_l],x_k][[x_i,x_j],x_k] \mod G_4$.