Prove that a Möbius transformation $T(z)=\dfrac{az+b}{cz+d}$ maps $\overline{\mathbb R}$ to $\overline{\mathbb R}$ if and only if it can be written with real coefficients.
If it can be written with real coefficients $a,b,c,d$, then for any $r \in \mathbb R, T(r)=\dfrac{ar+b}{cr+d}$, and by the closure property of $\mathbb R$ under addition and multiplication, we have $T(r) \in \mathbb R \cup \{\infty\}=\overline{\mathbb R}$
I don't know what to do to prove the other implication. I've tried to show it by the absurd: suppose $T$ maps $\overline{\mathbb R}$ to $\overline{\mathbb R}$ but $T$ can't be expressed with real coefficients, how could I arrive to an absurd? I would have to find $r \in \mathbb R$ such that $T(r) \in \mathbb C$. Maybe there is a straight forward way to prove it.
Hint: $T(0)$, $T(1)$, $T(-1)$ and $T(\infty)$ are all real.