“…From this viewpoint, it becomes interesting to find a reduction scheme relating the Lagrangian formulation of Palatini and Einstein-Hilbert gravity directly, without the detour through Hamiltonian formalism. So far, there exist two ways to implement reduction at the Lagrangian level, namely Lagrange-Poincaré reduction [8,11,12,17] and Routh reduction [4,6,14,18,31,32]. Moreover, there are physical considerations that can be said in support of this kind of reduction: They deal with not only the reduction, but also the reconstruction problem, and it is argued in [33] that reconstruction can be relevant from the physical point of view.…”