  We discuss the partitioning of a quantum system by subsystem separation
through unitary block-diagonalization (SSUB) applied to a Fock operator. Our
separation can be formulated in a very general way. It can be applied to very
different partitionings ranging from those driven by features in the molecular
structure (such as a solute surrounded by solvent molecules) to those that aim
at an orbital separation (such as core-valence separation). Our framework
embraces recent developments of Manby and Miller as well as older ones of
Huzinaga and Cantu. Projector-based embedding is simplified and accelerated by
SSUB. Moreover, it directly relates to decoupling approaches for relativistic
four-component many-electron theory. For a Fock operator based on the Dirac
one-electron Hamiltonian, one would like to separate the so-called positronic
(negative-energy) states from the electronic bound and continuum states. The
exact two-component (X2C) approach developed for this purpose becomes a special
case of the general SSUB framework and may therefore be viewed as a
system-environment decoupling approach. Moreover, for SSUB there exists no
restriction with respect to the number of subsystems that are generated - in
the limit, decoupling of all single-particle states is recovered, which
represents exact diagonalization of the problem. The fact that a Fock operator
depends on its eigenvectors poses challenges to all system-environment
decoupling approaches and is discussed in terms of the SSUB framework. Apart
from improved conceptual understanding, these relations bring about technical
advances as developments in different fields can immediately cross-fertilize
one another. As an important example we discuss the atomic decomposition of the
unitary block-diagonalization matrix in X2C-type approaches that can inspire
approaches for the efficient partitioning of large total systems based on SSUB.
