  We show that $\Gamma < \textbf{SU}(3,1)$ is a non-elementary complex
hyperbolic Kleinian group in which $tr(\gamma) \in \R$ for all $\gamma \in
\Gamma$ if and only if $\Gamma$ is conjugate to a subgroup of
$\textbf{SO}(3,1)$ or $\textbf{SU}(1,1)\times\textbf{SU}(2)$.
