  Recently, a new class of scalar constraint operators has been introduced in
loop quantum gravity. They are defined on a space of solutions to the Gauss
constraint and partial solutions to the vector constraint, called a vertex
Hilbert space. We propose a subspace of the vertex Hilbert space formed by
homogeneous-isotropic states, which is invariant under the action of the new
scalar constraint operators. As a result, the operators can be reduced to our
homogeneous-isotropic subspace. The (generalized) eigenstates of the reduced
operator are eigenstates of the full operator. We discuss the feasibility of
numerical diagonalization of the reduced scalar constraint operator.
