Orthogonal gauge fixing of first order gravity

We consider the first order connection formulation of 4D general relativity in the "orthogonal" gauge. We show how the partial gauge fixing of the phase space canonical coordinates leads to the appearance of second class constraints in the theory. We employ the gauge unfixing procedure in order to successfully complete the Dirac treatment of the system. While equivalent to the inversion of the Dirac matrix, the gauge unfixing allows us to work directly with the reduced phase space and the ordinary Poisson bracket. At the same time, we explicitly derive the new set of residual first class constraints preserving the partial gauge fixing, which are linear combinations of the original constraints, and these turn out to contain nonlinear terms. While providing an explicit example of how to consistently recast general relativity in a given partial gauge, the main motivation of this classical analysis is the application of the Quantum Reduced Loop Gravity program to a Schwarzschild black hole geometry.