  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.
