Even if an isolated defect results only in a local perturbation of the electron density, the wave function and the first-order reduced density matrix may still exhibit a long-range response to the defect. We present an axiomatic approach to the construction of a general-purpose embedding scheme which is able to cope with this problem. We start from a list of requirements, which we consider pertinent to an accurate embedding technique, and we proceed to demonstrate that the extended subspace approach recently proposed by Head and Silva [J. Chem. Phys. 104, 3244 (1996)] is the minimal realization of such an embedding scheme. The variational principle, strict fulfillment of the Pauli exclusion principle, a finite dimensional parameter space, and the possibility to perform the minimization by a standard SCF (self-consistent field) procedure are the key requirements which lead to a constrained SCF procedure. Self-embedding consistency and local completeness of the Hilbert space can then be realized by a mathematically very simple construction principle for the active subspace which can be formulated independent of any basis set. We analyze the spatial structure of the resulting minimal orbital space by means of tight-binding model Hamiltonians. For metal systems, we find active and frozen constrained SCF spaces to necessarily interlock in a strong and complicated fashion.