  The eight-vertex model on the square lattice with vertex weights $a,b,c,d$
obeying the relation $(a^2+ab)(b^2+ab)=(c^2+ab)(d^2+ab)$ is considered. Its
transfer matrix with $L=2n+1,\, n\geqslant 0,$ vertical lines and periodic
boundary conditions along the horizontal direction has the doubly-degenerate
eigenvalue $\Theta_n = (a+b)^{2n+1}$. A basis of the corresponding eigenspace
is investigated. Several scalar products involving the basis vectors are
computed in terms of a family of polynomials introduced by Rosengren and
Zinn-Justin. These scalar products are used to find explicit expressions for
particular entries of the vectors. The proofs of these results are based on the
generalisation of the eigenvalue problem for $\Theta_n$ to the inhomogeneous
eight-vertex model.
