  We have proposed previously a method for constructing self-conjugate
Hamiltonians H_eta in the eta-representation with a flat scalar product to
describe the dynamics of Dirac particles in arbitrary gravitational fields. In
this paper, we prove that, for block-diagonal metrics, the Hamiltonians H_eta
can be obtained, in particular, using "reduced" parts of Dirac Hamiltonians,
i.e. expressions for Dirac Hamiltonians derived using tetrad vectors in the
Schwinger gauge without or with a few summands with bispinor connectivities.
Based on these results, we propose a modified method for constructing
Hamiltonians in the eta-representation with a significantly smaller amount of
required calculations. Using this method, here we for the first time find
self-conjugate Hamiltonians for a number of metrics, including the Kerr metric
in the Boyer-Lindquist coordinates, the Eddington-Finkelstein,
Finkelstein-Lemaitre, Kruskal, Clifford torus metrics and for non-stationary
metrics of open and spatially flat Friedmann models.
