Corollary

Let us denote by $i : O^+ \to S^+/H$ and $d : S^+/H \to O^+$ the homomorphisms defined in Propositions 2 and 3, and let us write the abelian groups $O^+$ and $S^+/H$ additively; then $d(i(\theta)) = 2\theta$ for $\theta \in O^+$ and $i(d(\varphi)) = 2\varphi$ for $\varphi \in S^+/H$.

In fact, if $\overline{\varphi}$ is a rotation, we have $\overline{\varphi} = \varphi^{-1}$, whence $d(i(\varphi)) = \varphi/\overline{\varphi} = \varphi^2$. On the other hand, if $\varphi \in S^+/H$, $\varphi$ is the class mod. $H$ of a direct similarity $u$, and $d(\varphi) = u/\overline{u} = u^2 / N(u)$ is congruent to $u^2$ mod. $H$; whence the second formula.