Second derived subgroup

Definition with symbols
The second derived subgroup of a group $$G$$, denoted $$\! G''$$, is defined in the following equivalent ways:


 * 1) It is the subgroup generated by all elements of the form $$x,y],[z,w$$, where $$[,]$$ denotes the commutator and $$x,y,z,w \in G$$.
 * 2) It is the normal closure of the subgroup generated by all elements of the form $$x,y],[z,w$$ where $$[,]$$ denotes the commutator and $$x,y,z,w \in G$$.
 * 3) It is the defining ingredient::derived subgroup of the derived subgroup of $$G$$.
 * 4) It is the intersection of all subgroups $$H$$ of $$G$$ for which $$G/H$$ is a defining ingredient::metabelian group, i.e., a solvable group of derived length two.