Complex and Quaternionic hyperbolic Kleinian groups with real trace fields

Let $\mathrm{\Gamma }$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\mathrm{\Gamma }$ is contained in $\mathbb{R}$, $\mathrm{\Gamma }$ preserves a totally geodesic submanifold of constant negative sectional curvature. Furthermore if $\mathrm{\Gamma }$ is irreducible, $\mathrm{\Gamma }$ is a Zariski dense irreducible discrete subgroup of SO(n,1) up to conjugation. This is an analog of a theorem of Maskit for general semisimple Lie groups of rank $1$.