Lagrangian formulation of the Palatini action

We work on the Lagrangian formulation of the Palatini action. We find that we must assume the metric compatibility condition for the Palatini action to describe General Relativity, which condition should hold in quantization. We find that we must also assume one of the torsion zero condition or the tetrad compatibility condition. Our results will hold for any action in terms of the tetrad and the internal connection which describes General Relativity.