On Conjugation orbits of semisimple pairs in rank one

We consider Lie groups $\mathrm{S}\mathrm{U}(n,1)$ and $\mathrm{S}\mathrm{p}(n,1)$ that act as the isometries of the complex and quaternionic hyperbolic spaces respectively. We classify pairs of semisimple elements in $\mathrm{S}\mathrm{p}(n,1)$ and $\mathrm{S}\mathrm{U}(n,1)$ up to conjugacy. This gives local parametrization of the representations $\rho$ in $Hom(F_2,G)\mathrm{/}G$ such that both $\rho (x)$ and $\rho (y)$ are hyperbolics, where $F_2=⟨x,y⟩$, $G=\mathrm{S}\mathrm{p}(n,1)$ or $\mathrm{S}\mathrm{U}(n,1)$. We use the $\mathrm{P}\mathrm{S}\mathrm{p}(n,1)$-configuration space $M(n,i,m-i)$ of ordered $m$-tuples of points on $\overline{{\mathbf{H}}_{\mathbb{H}}^n}$, where first $i$ points in an $m$-tuple are null points, to classify the semisimple pairs. Further, we also classify points on $M(n,i,m-i)$.