  In the popular ${\cal PT}-$symmetry-based formulation of quantum mechanics of
closed systems one can build unitary models using non-Hermitian Hamiltonians
(i.e., $H \neq H^\dagger$) which are Hermitizable (so that one can write,
simultaneously, $H = H^\ddagger$). The essence of the trick is that the
reference Hilbert space $\cal R$ (in which we use the conventional inner
product $\langle \psi_a|\psi_b\rangle$ and write $H \neq H^\dagger$) is
declared unphysical. The necessary Hermiticity of the Hamiltonian $H =
H^\ddagger$ can be then achieved by the mere metric-mediated amendment $\langle
\psi_a|\Theta|\psi_b\rangle$ to the inner product. This converts $\cal R$ into
a correct physical Hilbert space $\cal H$. The feasibility of the construction
is based on a factorization postulate $\Theta={\cal PC}$ where, usually, ${\cal
P}$ is parity and ${\cal C}$ is charge. In our paper we propose a more general
factorization recipe in which one constructs $\Theta=Z_NZ_{N-1}\ldots Z_1$, at
any $N$, in terms of suitable auxiliary pre-metric operators $Z_k$.
