  The current paper is dedicated to developing a (3+1) decomposition for the
minimal gravitational Standard-Model Extension. Our setting is explicit
diffeomorphism violation and we focus on the background fields known in the
literature as $u$ and $s^{\mu\nu}$. The Hamiltonian formalism is developed for
these contributions, which amounts to deriving modified Hamiltonian and
momentum constraints. We then study the connection between these modified
constraints and the modified Einstein equations. Implications are drawn on the
form of the background fields to guarantee the internal consistency of the
corresponding modified-gravity theories. In the course of our analysis, we
obtain a set of consistency requirements for $u$ and certain sectors of
$s^{\mu\nu}$. We argue that the constraint structure remains untouched when
these conditions are satisfied. Our results shed light on explicit violations
of diffeomorphism invariance and local Lorentz invariance in gravity. They may
turn out to be valuable for developing a better understanding of effective
modified-gravity theories.
