Let $G$ be $SL_2(\mathbb{R})$, the groups of real $2 \times 2$ matrices of determinant $1$, acting on $\mathbb{C}\cup \infty$ by M¨obius transformations. For each of the points $0$, $i$, $−i$, compute its stabilizer and its orbit under the action of $G$. Show that $G$ has exactly $3$ orbits in all.
I have computed their stabilizer and orbits but not sure how to show $G$ only has exactly three orbits. Greatly appreciated if anyone can help!