Let $F$ denote the free group on $n$ generators $g\_1,\ldots, g\_n$. Consider its quotient $Q$ by the universal relation $[x,[x,y]]$ (a "Serre relation" familiar from Lie theory). This group is nilpotent of class $\leq n$. Denote by $Q\_k$ the k-th term of its lower central series. It appears that all commutators of length $\geq 3$ are torsion elements in $Q$, while $Q\_2/Q\_3$ is the abelian group freely generated by $\{ [g\_i, g\_j], i<j\}$. Is the precise structure of $Q$ known?



A nilpotent quotient of free groups
As [Ian Agol](https://mathoverflow.net/users/1345/ian-agol) indicated in his comment, groups such as $Q$ have been studied for a while. What you wrote is a specific case of a [$n$-Engel group](http://en.wikipedia.org/wiki/Engel_group) with $n = 2$. The case $n=2$ is, for the most part, completely understood. See this [page](http://groupprops.subwiki.org/wiki/2-Engel_group) from group props. Also check out Levi's article *Groups in which the commutator operation satisfies certain algebraic conditions* (J. Indian Math. Soc. (N.S.) 6, (1942). 87–97). Here it is shown that $Q$ has nilpotence class at most $3$. See also Burnside, W.; *On Linear Homogeneous Continuous Groups whose Operations are Permutable.* (Proc. London Math. Soc. S1-29 no. 1, 325).


For a start on general $n$, see Havas, George; Vaughan-Lee, M. R. *4-Engel groups are locally nilpotent* (Internat. J. Algebra Comput. 15 (2005), no. 4, 649–682).


