Let $F$ be a free group of letters $\{x_1,...,x_m\}$ and define $$F_1:=F$$ $$F_k:=[F_{k-1},F]$$ According to this question, $F_2/F_3$ and $F_3/F_4$ are generated by representatives given by basic commutators in $F_2$ and $F_3$ respectively, i.e. $[x_i,x_j]$ and $[[x_i,x_j],x_k]$.
Is $F_k/F_{k+1}$ generated by basic commutators in $F_k$ ?
By definition any element can be written as a product of commutators which are not necessarily the basic commutators.
If a proof is too technical I would accept an answer stating the result and possibly a source where one can look for the proof itself.