  The analysis of the transfer matrices associated to the most general
representations of the 8-vertex reflection algebra on spin-1/2 chains is here
implemented by introducing a quantum separation of variables (SOV) method which
generalizes to these integrable quantum models the method first introduced by
Sklyanin. More in detail, for the representations reproducing in their
homogeneous limits the open XYZ spin-1/2 quantum chains with the most general
integrable boundary conditions, we explicitly construct representations of the
8-vertex reflection algebras for which the transfer matrix spectral problem is
separated. Then, in these SOV representations we get the complete
characterization of the transfer matrix spectrum (eigenvalues and eigenstates)
and its non-degeneracy. Moreover, we present the first fundamental step toward
the characterization of the dynamics of these models by deriving determinant
formulae for the matrix elements of the identity on separated states, which
apply in particular to transfer matrix eigenstates. The comparison of our
analysis for the 8-vertex reflection algebra with that of [1, 2] for the
6-vertex one leads to the interesting remark that a profound similarity in both
the characterization of the spectral problems and of the scalar products exists
for these two different realizations of the reflection algebra once they are
described by SOV method. As it will be shown in a future publication, this
remarkable similarity will be at the basis of the simultaneous determination of
form factors of local operators of integrable quantum models associated to
general reflection algebra representations of both 8-vertex and 6-vertex type.
