Local coordinates for complex and quaternionic hyperbolic pairs

Let $G(n)=\mathrm{S}\mathrm{p}(n,1)$ or $\mathrm{S}\mathrm{U}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$. Such pairs, called non-singular', were introduced by Gongopadhyay and Parsad for ${\rm SU}(3,1)$. We extend this notion and classify $G(n)$-conjugation orbits of such elements in arbitrary dimension. We prove that the set given by non-singular pairs in $G(n)$ is small’ for $n\ge 4$. However, for $n=3$, they give a subspace that can be parametrized using a set of coordinates whose local dimension equals the dimension of the underlying group. We further construct twist-bend parameters to glue such representations and obtain local parametrization for generic representations of the fundamental group of a closed oriented surface into $G(3)$.