Marginal subgroup for variety of metabelian groups

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
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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 centralizer of derived subgroup $C_{G}([G,G])$ and such that the quotient group $H/C_{G}([G,G])$ is the 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 marginal subgroup corresponding to the variety of metabelian groups.

Relation with other subgroup-defining functions

Smaller subgroup-defining functions

Subgroup-defining function	Meaning
center	commutes with all elements
second center	second member of upper central series, also, marginal subgroup for nilpotency class two
centralizer of derived subgroup	centralizer of derived subgroup
third center	third member of upper central series, also, marginal subgroup for nilpotency class three.