  It is known that the dynamics of dissipative fluids in Eulerian variables can
be derived from an algebra of Leibniz brackets of observables, the metriplectic
algebra, that extends the Poisson algebra of the zero viscosity limit via a
symmetric, semidefinite component. This metric bracket generates dissipative
forces.
  The metriplectic algebra includes the conserved total Hamiltonian $H$,
generating the non-dissipative part of dynamics, and the entropy $S$ of those
microscopic degrees of freedom draining energy irreversibly, that generates
dissipation. This $S$ is a Casimir of the Poisson algebra to which the
metriplectic algebra reduces in the frictionless limit.
  In the present paper, the metriplectic framework for viscous fluids is
re-written in the Lagrangian Formulation, where the system is described through
material variables: this is a way to describe the continuum much closer to the
discrete system dynamics than the Eulerian fields. Accordingly, the full
metriplectic algebra is constructed in material variables, and this will render
it possible to apply it in all those cases in which the Lagrangian Formulation
is preferred.
  The role of the entropy $S$ of a metriplectic system is as paramount as that
of the Hamiltonian $H$, but this fact may be underestimated in the Eulerian
formulation because $S$ is not the only Casimir of the symplectic non-canonical
part of the algebra. Instead, when the dynamics of the non-ideal fluid is
written through the parcel variables of the Lagrangian formulation, the fact
that entropy is symplectically invariant appears to be more clearly related to
its dependence on the microscopic degrees of freedom of the fluid, that do not
participate at all to the symplectic canonical part of the algebra (which,
indeed, involves and evolves only the macroscopic degrees of freedom of the
fluid parcel).
