  This work outlines a consistent method of identifying subsystems in
finite-dimensional Hilbert spaces, independent of the underlying inner-product
structure. It has been well established that Hilbert spaces with modified
inner-product, defined through the so-called metric operator, turn out to be
the most natural ways to represent certain phenomena such as those involving
balanced gain and loss resulting in pseudo-Hermitian Hamiltonians. For
composite systems undergoing pseudo-Hermitian evolution, defining the
subsystems is generally considered feasible only when the metric operator is
chosen to have a tensor product form so that a partial trace operation can be
well defined. In this work, we use arguments from algebraic quantum mechanics
to show that the subsystems can be well-defined in every metric space --
irrespective of whether or not the metric is of tensor product form. This is
done by identifying subsystems with a decomposition of the underlying
$C^*$-algebra into commuting sub-algebras. We show that different subsystem
decompositions correspond to choosing different equivalence classes of the GNS
representation. Furthermore, given a form of pseudo-Hermitian Hamiltonian, the
choice of the Hamiltonian compatible metric characterizes the subsystem
decomposition and as a consequence, the entanglement structure in the system.
We clarify how each of the subsystems, defined this way, can be tomographically
constructed and that these subsystems satisfy the no-signaling principle. With
these results, we put all the choices of the metric operator on an equal
footing.
