G-invariance of set of points fixed by loxodromic elements in G

Question

Let G be a non-elementary subgroup of Mobius transformations. How can we show that the set of points fixed by loxodromic elements in G is G-invariant?

I proved it by directly computations, but I would like to ask for other ways to show it.

Answer 1

Suppose $x$ is fixed by a loxodromic element $g \in G$. Consider $h \in G$ and $y=h(x)$. An easy computation shows that $y$ is fixed by $hgh^{-1} \in G$: $$hgh^{-1}(y)=hg(x)=h(x)=y $$ Also, using that $g$ is loxodromic one can prove that $hgh^{-1}$ is loxodromic (there are several proofs, all are easy, but I will hold off giving one since I do not know what definition of loxodromic you re using).

One thing to note: the hypothesis that $G$ be non-elementary is not needed for this proof.