Marginal subgroup for variety of metabelian groups

Definition
Let $$G$$ be a group. The marginal subgroup for variety of metabelian groups for $$G$$ is defined in the following equivalent ways:


 * 1) It is a subgroup $$H$$ that contains the defining ingredient::centralizer of derived subgroup $$C_G([G,G])$$ and such that the quotient group $$H/C_G([G,G])$$ is the defining ingredient::center of the quotient group $$G/C_G([G,G])$$.
 * 2) It is precisely the set of elements $$g \in G$$ such that for any (possibly equal or unequal) $$x_1,x_2,x_3,x_4 \in G$$, we have $$x_1,x_2],[x_3,x_4 = gx_1,x_2],[x_3,x_4$$.
 * 3) It is precisely the set of elements $$g \in G$$ such that for any (possibly equal or unequal) $$x_1,x_2,x_3,x_4 \in G$$, we have $$x_1,x_2],[x_3,x_4 = gx_1,x_2],[x_3,x_4 = x_1,gx_2],[x_3,x_4 =x_1,x_2],[gx_3,x_4 =x_1,x_2],[x_3,gx_4$$.
 * 4) It is the defining ingredient::marginal subgroup corresponding to the variety of defining ingredient::metabelian groups.