We use the definitions of this question.
Is the following proposition true? If yes, how do we prove it?
Proposition Let $D$ be a non-square integer such that $D \equiv 0$ (mod $4$) or $D \equiv 1$ (mod $4$). There exists a bijection $\psi\colon Cl^+(R) \rightarrow C(D)$ by the proposition of this question. We identify $C(D)$ with $Cl^+(R)$ by $\psi$. Hence $C(D)$ is an abelian group with this identification. Let $[F]$ be a class of $C(D)$ represented by a primitive form $F = ax^2 + bxy + cy^2$. Let $G = ax^2 - bxy + cy^2$. Then $[F][G] = 1$.