I've been trying to understand geodesics of hyperbolic space $H^n$, and found a very similar question and a helpful answer here: Geodesics and Distance in Hyperbolic Space
However, I found it difficult to prove one point in the 2nd step of the above-mentioned page: "Show there is an element of $O_+(1,n)$ fixing $P$ and acting non-trivially on each point of $\mathbb{R}^{n+1}\backslash P$."
How can I find such $A \in O_+(1,n)$? Thanks in advance.