Group of nilpotency class two powered over a commutative unital ring

Definition
Suppose $$R$$ is a ring that is commutative, associative and unital. Suppose $$G$$ is a group of nilpotency class two. The structure of $$G$$ as a group powered over $$R$$ is a mapping:

$$G \times R \to G$$

written using exponentiation, i.e., the image of $$(g,r)$$ is written $$g^r$$, satisfying the following conditions:


 * 1) Agreement on integers with usual powering: Denote by $$\iota:\mathbb{Z} \to R$$ the natural homomorphism that sends 1 to 1. Then, for any $$n \in \mathbb{Z}$$, we have $$g^n = g^{\iota(n)}$$. (Actually, in light of the other axioms, it suffices to assume that $$g^1 = g$$, the rest follows).
 * 2) Exponentials turn multiplication to addition: We have $$g^rg^s = g^{r + s}$$ for all $$g \in G, r,s \in R$$
 * 3) Composition of exponentials is exponential by product: We have $$(g^r)^s = g^{rs}$$ for all $$g \in G, r,s \in R$$.
 * 4) Class two condition intended to mimic the formula for powers of product in group of class two:

$$x^ry^r = [x,y]^s(xy)^r$$

for all $$x,y \in G, r,s \in R$$ for which $$2s= r(r - 1)$$. Here $$[x,y]$$ denotes the commutator of $$x$$ and $$y$$. Because the group has class two, it does not matter whether we use the left normed convention or the right normed convention for the commutator.