Group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two

Definition
A group of nilpotency class two whose commutator map is the double of an alternating bihomomorphism giving class two is a group $$G$$ satisfying both the following:


 * 1) $$G$$ is a defining ingredient::group of nilpotency class two.
 * 2) There is a function $$\circ :  G \times G \to G$$ such that $$(x \circ y)^2 = [x,y]$$ for all $$x,y \in G$$, where $$[, ]$$ denotes the commutator in the group, $$\circ$$ is a bihomomorphism, $$x \circ x $$ is the identity element and $$x \circ y = (y \circ x)^{-1}$$ for all $$x,y \in G$$ and $$(x \circ y) \circ z$$ is the identity element for all $$x,y,z \in G$$.

Note that for class two, the left and right conventions for commutator coincide, so it does not matter which one we pick.