  It is well known that a theory with explicit Lorentz violation is not
invariant under diffeomorphisms. On the other hand, for geometrical theories of
gravity, there are alternative transformations, which can be best defined
within the first-order formalism, and that can be regarded as a set of improved
diffeomorphisms. These symmetries are known as local translations and, among
other features, they are Lorentz covariant off shell. It is thus interesting to
study if theories with explicit Lorentz violation are invariant under local
translations. In this work, an example of such a theory, known as the minimal
gravity sector of the Standard Model Extension, is analyzed. Using a robust
algorithm, it is shown that local translations are not a symmetry of the
theory. It remains to be seen if local translations are spontaneously broken
under spontaneous Lorentz violation, which are regarded as a more natural
alternative when spacetime is dynamic.
