Exercise 7.5.9

Proof. $\text{∙}$ The text proves that the isomorphism $r\mapsto [\gamma ]$ sends $r_1=\mathrm{Rot}(\pi ,\stackrel{\to }{e_1})$ on ${\gamma }_1$, where

$\text{∙}$ Let ${\gamma }_2=\left(\begin{array}{cc}\hfill i\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)\in \mathrm{GL}(2,F)$. The homography $[{\gamma }_2]$ satisfies $[{\gamma }_2]\cdot z=\mathit{iz}$ for all $z\in ℂ$, and $[{\gamma }_2]\cdot \infty =\infty$. So

The rotation $r_2=\mathrm{Rot}(\pi ∕2,\stackrel{\to }{e_3})$ satisfies

Thus

By the uniqueness proved in Exercise 8, $[\hat{\pi }\circ r_2\circ {\hat{\pi }}^{-1}]=[{\gamma }_2]$.

In other words, the isomorphism $r\mapsto [\gamma ]$ sends $r_2=\mathrm{Rot}(\pi ∕2,\stackrel{\to }{e_3})$ on ${\gamma }_2$, where

$\text{∙}$ Let ${\gamma }_3=\left(\begin{array}{cc}\hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 1\hfill \end{array}\right)\in \mathrm{GL}(2,F)$.

The homography $[{\gamma }_3]$ satisfies $[{\gamma }_3]\cdot z=\frac{z-1}{z+1}$ for all $z\in ℂ$, and $[{\gamma }_3]\cdot \infty =1$. So

The rotation $r_3=\mathrm{Rot}(\pi ∕2,\stackrel{\to }{e_2})$ satisfies

thus

By the same uniqueness property, $[\hat{\pi }\circ r_3\circ {\hat{\pi }}^{-1}]=[{\gamma }_3]$.

In other words, the isomorphism $r\mapsto [\gamma ]$ sends $r_3=\mathrm{Rot}(\pi ∕2,\stackrel{\to }{e_3})$ on ${\gamma }_3$, where

□