  In paper [Znojil M., Phys. Rev. D 78 (2008), 085003, 5 pages,
arXiv:0809.2874] the two-Hilbert-space (2HS, a.k.a. cryptohermitian)
formulation of Quantum Mechanics has been revisited. In the present
continuation of this study (with the spaces in question denoted as ${\cal
H}^{\rm (auxiliary)}$ and ${\cal H}^{\rm (standard)}$) we spot a weak point of
the 2HS formalism which lies in the double role played by ${\cal H}^{\rm
(auxiliary)}$. As long as this confluence of roles may (and did!) lead to
confusion in the literature, we propose an amended, three-Hilbert-space (3HS)
reformulation of the same theory. As a byproduct of our analysis of the
formalism we offer an amendment of the Dirac's bra-ket notation and we also
show how its use clarifies the concept of covariance in time-dependent cases.
Via an elementary example we finally explain why in certain quantum systems the
generator $H_{\rm (gen)}$ of the time-evolution of the wave functions may
differ from their Hamiltonian $H$.
