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We show that $\Gamma < \textbf{SU}(3,1)$ is a non-elementary complexhyperbolic 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 $\textbf{SU}(1,1)\times\textbf{SU}(2)$.

A characterization of complex hyperbolic Kleinian groups in dimension 3 with trace fields contained in $\mathbb R$