  For a generic overdamped Langevin dynamics driven out of equilibrium by both
time-dependent and nonconservative forces, the entropy production rate can be
decomposed into two positive terms, termed excess and housekeeping entropy.
However, this decomposition is not unique: There are two distinct
decompositions, one due to Hatano and Sasa, the other one due to Maes and
Neto\v{c}ny. Here, we establish the connection between these two decompositions
and provide a simple, geometric interpretation. We show that this leads to a
decomposition of the entropy production rate into three positive terms, which
we call excess, housekeeping and coupling part, respectively. The coupling part
characterizes the interplay between the time-dependent and nonconservative
forces. We also derive thermodynamic uncertainty relations for the excess and
housekeeping entropy in both the Hatano-Sasa and Maes-Neto\v{c}ny-decomposition
and show that all quantities obey integral fluctuation theorems. We illustrate
the decomposition into three terms using a solvable example of a dragged
particle in a nonconservative force field.
