Consistent and inconsistent truncations. Some results and the issue of the correct uplifting of solutions

We clarify the existence of two different types of truncations of the field content in a theory, the consistency of each type being achieved by different means. A proof is given of the conditions to have a consistent truncation in the case of dimensional reductions induced by independent Killing vectors. We explain in what sense the tracelessness condition found by Scherk and Scharwz is not only a necessary condition but also a {\it sufficient} one for a consistent truncation. The reduction of the gauge group is fully performed showing the existence of a sector of rigid symmetries. We show that truncations originated by the introduction of constraints will in general be inconsistent, but this fact does not prevent the possibility of correct upliftings of solutions in some cases. The presence of constraints has dynamical consequences that turn out to play a fundamental role in the correctness of the uplifting procedure.