I want to understand Prop 6 in the paper "Convergence of the Q-curvarture flow on $S^4$" by Simon Brendle. I understand that for every $p\in B^4$, where $B^4=\{x\in\mathbb{R}^5: |x|\leq 1\}$, $$\phi(x)=p+\frac{1-|p|^2}{1+2\langle p,x\rangle+|p|^2}(x+p) $$defines a conformal diffeomorphism from $S^4$ to $S^4$, where $S^4=\{x\in\mathbb{R}^5: |x|=1\}$ is the sphere (This fact was proved in the paper in P.11). My questions are:

(i) How is the formula derived? I want to understand how can we know the conformal map is related to some $p\in B^4$ and why the formula is in this particular form. Of course, if I already know the formula I can just check by definition it is a conformal map.

(ii) Is every conformal diffeomorphism from $S^4$ to $S^4$ in this form? I think this is related to Liouville's Theorem.