Hello everyone!

In a paper "Coverings of twisted Chevalley groups over commutative rings" Eiichi Abe intoroduced the following construction:

Let $R$ be a commutative ring and $x\mapsto\overline{x}$ an involution on it. Set $$A=\left\{(a,b)\ |\ a\overline{a}=b+\overline{b}\right\}\subset R\times R.$$ The set $A$ has binary operation on it, namely $(a,b)\cdot(c,d)=(a+c,b+d+\overline{a}c)$.

This is connected with what's called Heisenberg group $H$ of a form $B$, where $B$ is a sesquilinear antihermitian form on a right $R$-module $V$. As a set, it coincides with $V\times R$, an operation is defined by $(u,a)\cdot(v,b)=(u+v,a+b-B(u,v))$. Abe's group $A$ coincides with so-called maximal form parameter, that is $$\Lambda_{max}=\left\{ \xi\in H\ |\ tr(\xi)=0\right\}$$ where $tr((u,a))=a-\overline{a}+B(u,u)$.

I'm interested if those groups has been studied by someone, at least over some classes of fields? In case of finite field with non-trivial involution and complex numbers it's easy to study them, but in other cases it isn't. I tried to google it, but I failed.