A characterization of quaternionic Kleinian groups in dimension 2 with   complex trace fields
  Let $G$ be a non-elementary discrete subgroup of $\mathrm{Sp}(2,1)$. We show that if the sum of diagonal entries of each element of $G$ is a complex number, then $G$ is conjugate to a subgroup of $\mathrm{U}(2,1)$. 
Sungwoon Kim, Joonhyung Kim
math.GT