  Diffeomorphisms and an internal symmetry (e.g., local Lorentz invariance) are
typically regarded as the symmetries of any geometrical gravity theory,
including general relativity. In the first-order formalism, diffeomorphisms can
be thought of as a derived symmetry from the so-called local translations,
which have improved properties. In this work, the algebra of an arbitrary
internal symmetry and the local translations is obtained for a generic gauge
theory of gravity, in any spacetime dimensions, and coupled to matter fields.
It is shown that this algebra closes off shell suggesting that these symmetries
form a larger gauge symmetry. In addition, a mechanism to find the symmetries
of theories that have nondynamical fields is proposed. It turns out that the
explicit form of the local translations depend on the internal symmetry and
that the algebra of local translations and the internal group still closes off
shell. As an example, the unimodular Einstein-Cartan theory in four spacetime
dimensions, which is only invariant under volume preserving diffeomorphisms, is
studied.
