  In this paper we elevate the formalism for off-shell Abbott-Deser-Tekin (ADT)
conserved charges in the presence of arbitrary matter fields to including the
internal gauge transformation, when the gauge fields are present. For this
purpose, we resort to exact symmetry, and the symmetry generator is combined by
diffeomorphism and internal gauge transformation. For universality, we consider
an apparently non-covariant Lagrangian, which contains a matter Chern-Simons
term. We show that the elevated off-shell ADT formalism is equivalent to the
covariant phase space method and the Barnich-Brandt-Comp\`ere (BBC) formalism.
To check the validity of our method, we explicitly compute the conserved
charges of general non-extremal rotating charged G\"odel black holes in minimal
five-dimensional gauged supergravity, reproducing the previously known results.
