Conjugation Orbits of Loxodromic Pairs in SU(n,1)

Let ${bf H}_{mathbb C}^n$ be the $n$-dimensional complex hyperbolic space and ${rm SU}(n,1)$ be the (holomorphic) isometry group. An element $g$ in ${rm SU}(n,1)$ is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary $partial {bf H}_{mathbb C}^n$. We classify ${rm SU}(n,1)$ conjugation orbits of pairs of loxodromic elements in ${rm SU}(n,1)$.