  We derive an exact operatorial reformulation of the rotational invariant
slave boson method and we apply it to describe the orbital differentiation in
strongly correlated electron systems starting from first principles. The
approach enables us to treat strong electron correlations, spin-orbit coupling
and crystal field splittings on the same footing by exploiting the gauge
invariance of the mean-field equations. We apply our theory to the archetypical
nuclear fuel UO$_2$, and show that the ground state of this system displays a
pronounced orbital differention within the $5f$ manifold, with Mott localized
$\Gamma_8$ and extended $\Gamma_7$ electrons.
