Physical dynamical systems amount to placing vector fields on manifolds. Important examples are noncanonical Hamiltonian systems, i.e., where the manifold is a Poisson manifold and the vector field is generated by a Hamiltonian function and a degenerate Poisson bracket. Physical systems often have vector fields composed of the sum of a Hamiltonian part and a dissipative part, that may or may not be thermodynamically consistent. Metriplectic dynamics is a kind of dynamical system (finite or infinite) that ensures thermodynamic consistency: conservation of energy and production of entropy. Metriplectic flows exist on Poisson manifolds that posses an additional structure, the metriplectic 4-bracket that maps four phase space functions to another, and has algebraic curvature symmetry. A consequence of this structure is the identification of the rate of entropy production as sectional curvature defined by the level sets of energy (the Hamiltonian) and entropy (a Casimir invariant). Metriplectic 4-brackets can be constructed using the Kulkarni-Nomizu product or via a pure Lie algebraic formalism based on the Koszul connection. The formalism produces many known and new dynamical systems.