Real determinant-one Möbius orbits ($\mathbb{R}\cup \{\infty \},\mathrm{Im}z>0,\mathrm{Im}z<0$)

The special linear group $S{L}_2(\mathbb{R})$ acting by Möbius transformations has exactly three group orbits on the Riemann sphere: the extended real line and the two open half-planes. The identity $\mathrm{Im}((az+b)/(cz+d))=\mathrm{Im}z/|cz+d{|}^2$ preserves these sets. Upper triangular determinant-one matrices send $i$ to any prescribed point of the complex upper half-plane, and similarly send $-i$ through the lower half-plane. The stabilizer subgroups of $i$ and $-i$ are both $SO(2)$.