After googling a bit I was surprised that I couldn't find any reference in which the action of $\mathrm{SL}(2,\mathbb R)$ on the Poincaré disk corresponding to the usual action by Möbius transformations on the upper half plane is computed explicitly. Of course I could do it by hand but it is a little bit tedious and I am sure this is written in some textbook.
What I am precisely looking for is an explicit, purely real formula for $y_1$ and $y_2$ in terms of $a,b,c,d,x_1,x_2$ when $x=(x_1,x_2)$ lies in the open unit disk, $g\in \mathrm{SL}(2,\mathbb R)$ has entries $a,b,c,d$ and $$y=(y_1,y_2)=g\cdot x.$$
Make $\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{SL}_2(\mathbb R)$ act as $$z\mapsto \frac{((a+d)+(b-c)i)z+((b+c)+(a-d)i)}{((b+c)-(a-d)i)z+((a+d)-(b-c)i)}.$$