Conjugation orbits of loxodromic pairs in SU(n,1)

Let HCn be the n-dimensional complex hyperbolic space and SU(n,1) be the (holomorphic) isometry group. An element g in SU(n,1) is called loxodromic or hyperbolic if it has exactly two fixed points on the boundary ∂HCn. We classify SU(n,1) conjugation orbits of pairs of loxodromic elements in SU(n,1).