Consistent Pauli Sphere Reductions and the Action
  It is a commonly held belief that a consistent dimensional reduction ansatz can be equally well substituted into either the higher-dimensional equations of motion or the higher-dimensional action, and that the resulting lower-dimensional theories will be the same. This is certainly true for Kaluza-Klein circle reductions and for DeWitt group-manifold reductions, where group-invariance arguments guarantee the equivalence. In this paper we address the question in the case of the non-trivial consistent Pauli coset reductions, such as the S^7 and S^4 reductions of eleven-dimensional supergravity. These always work at the level of the equations of motion. In some cases the reduction ansatz can only be given at the level of field strengths, rather than the gauge potentials which are the fundamental fields in the action, and so in such cases there is certainly no question of being able to substitute instead into the action. By examining explicit examples, we show that even in cases where the ansatz can be given for the fundamental fields appearing in an action, substituting it into the higher-dimensional action may not give the correct lower-dimensional theory. This highlights the fact that much remains to be understood about the way in which Pauli reductions work. 
H. Lu, C.N. Pope and K.S. Stelle