A characterization of quaternionic Kleinian groups in dimension 2 with complex trace fields

Let $G$ be a non-elementary discrete subgroup of $\mathrm{S}\mathrm{p}(2,1)$. 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 $\mathrm{U}(2,1)$.