  Let $\Sigma_g$ be a compact, connected, orientable surface of genus $g \geq
2$. We ask for a parametrization of the discrete, faithful, totally loxodromic
representations in the deformation space ${\rm Hom}(\pi_1(\Sigma_g), {\rm
SU}(3,1))/{\rm SU}(3,1)$. We show that such a representation, under some
hypothesis, can be determined by $30g-30$ real parameters.
