I have seen many solutions which are very intricate and/or long. The solution I had was much shorter which lead me to believe that it was incorrect. It went like this :
$T$ be a Mobius transformation such that $T(\Bbb R_∞)$=$\Bbb R_∞$. $1,0,∞\in \Bbb R_∞$ so their inverse images are also in $\Bbb R_∞$. $(z,z_2,z_3,z_4)$ is the unique Mobius transformation that maps $z_2,z_3,z_4$ to respectively $1,0,∞$. So when we write $T$ as a cross-ratio, the coefficients are all real.
Is this solution really incorrect?