Commutator-verbal subgroup

Definition
A subgroup of a group is a commutator-verbal subgroup if there is a set of words, each having the format of a commutator involving its letters and their inverses, such that the subgroup is precisely the subgroup generated by all the elements of the group expressible using those words.

Examples

 * All members of the derived series are commutator-verbal subgroups.
 * All members of the lower central series are commutator-verbal subgroups.
 * The subgroup generated by all elements of the form $$[[x,y],y]$$ is a commutator-verbal subgroup.
 * For a group $$G$$, the subgroup $$G,G],[[G,G],G$$ is a commutator-verbal subgroup.