I wanna show that if $A \in PSL(2,\mathbb{C})$ then $A= i_{c_1} \circ i_{c_2}$ with $i_{c_j}$ inversion of the sphere $c_j$.
I tried to show it so, I can to identify $A$ with $\frac{az+b}{cz+d}$ through the projection action, then I know that $i_c$ has form $i(x) = a + \frac{r^2}{|x-a|^2}(x-a)$, for a circle $c=C(a,r)$ with $a$ center and $r$ radius, I composed two of it but I have not obtained good news.
Other thing that i know is:
- $A$ has one or two fixed points
- $\frac{az+b}{cz+d}$ send generalized circles in circles or lines
I have already shown it
Could you give me some hint?
Thanks