  A variational formulation for nonequilibrium thermodynamics was recently
proposed in \cite{GBYo2017a,GBYo2017b} for both discrete and continuum systems.
This formulation extends the Hamilton principle of classical mechanics to
include irreversible processes. In this paper, we show that this variational
formulation yields a constructive and systematic way to derive from a unified
perspective several bracket formulations for nonequilibrium thermodynamics
proposed earlier in the literature, such as the single generator bracket and
the double generator bracket. In the case of a linear relation between the
thermodynamic fluxes and the thermodynamic forces, the metriplectic or GENERIC
bracket is recovered. We also show how the processes of reduction by symmetry
can be applied to these brackets. In the reduced setting, we also consider the
case in which the coadjoint orbits are preserved and explain the link with
double bracket dissipation. A similar development has been presented for
continuum systems in \cite{ElGB2019} and applied to multicomponent fluids.
