  We apply the converse of Noether's second theorem to the first-order
$n$-dimensional Lovelock action, considering the frame rotation group as both
$SO\left(1,n-1\right)$ or as $SO(n)$. As a result, we get the well-known
invariance under local Lorentz transformations or $SO(n)$ transformations and
diffeomorphisms, for odd- and even-dimensional manifolds. We also obtain the
so-called `local translations' with nonvanishing constant $\Lambda$ for
odd-dimensional manifolds when a certain relation among the coefficients of the
various terms of the first-order Lovelock Lagrangian is satisfied. When this
relation is fulfilled, we report the existence of a new gauge symmetry emerging
from a Noether identity. In this case the fundamental set of gauge symmetries
of the Lovelock action is composed by the new symmetry, local translations with
$\Lambda \neq 0$ and local Lorentz transformations or $SO(n)$ transformations.
The commutator algebra of this set closes with structure functions. We also get
the invariance under local translations with $\Lambda=0$ of the highest term of
the Lovelock action in odd-dimensional manifolds. Furthermore, we report a new
gauge symmetry for the highest term of the first-order Lovelock action for
odd-dimensional manifolds. In this last case, the fundamental set of gauge
symmetries can be considered as Poincar\'e or Euclidean transformations
together with the new symmetry. The commutator algebra of this set also closes
with structure functions.
