Recent work in 3D Pose Graph Optimization (PGO) shows how a dual Lagrangian formulation of the problem can be used to verify (and possibly certify) the quality of a given solution. A limitation of current approaches is that they relax the positive determinant constraint for the rotations: in consequence, if they fail to certify an optimal solution (i.e., if the duality gap is nonzero), one cannot determine if this is due to the relaxation itself, or if it is an intrinsic feature of the problem at hand. In this paper we show how one can include the determinant constraints in the derivation of the dual, thus showing that their relaxation is unnecessary. We show experimentally that this complete formulation does not lead to tangible differences with respect to the original, relaxed version. This indicates that the reasons for failure in providing a certificate of optimality are intrinsic to the problem, and that the determinant constraints are somehow redundant in common PGO instances.