If $T$ is a Möbius transformation with $ Tz = \frac{az+b}{cz+d} $ and $ T(\mathbb{R}_{\infty}) = \mathbb{R}_{\infty} $, show that we can choose $a,b,c,d$ to be real numbers.
This is an exercise from Conway.
Now, $ T(\infty) = {a \over c} \in \mathbb{R}, T(0) = { b \over d} \in \mathbb{R} $ so $ a$ and $b$ are real multiple of $c$ and $d$, but how does this implies we can choose them all real?
Möbius transformations are bijective mappings of the extended complex plane onto itself, and therefore $T(\mathbb{R}_{\infty}) = \mathbb{R}_{\infty}$ implies also that $$ T^{-1}(\infty) = -\frac dc \in \mathbb{R} \, . $$ (If $c = 0$ then consider $T^{-1}(0)$ instead.)