  The authors previously introduced a diffeomorphism-invariant definition of a
homogeneous and isotropic sector of loop quantum gravity, along with a program
to embed loop quantum cosmology into it. The present paper works out that
program in detail for the simpler, but still physically non-trivial, case where
the target of the embedding is the homogeneous, but not isotropic, Bianchi I
model. The diffeomorphism-invariant conditions imposing homogeneity and
isotropy in the full theory reduce to conditions imposing isotropy on an
already homogeneous Bianchi I spacetime. The reduced conditions are invariant
under the residual diffeomorphisms still allowed after gauge fixing the Bianchi
I model. We show that there is a unique embedding of the quantum isotropic
model into the homogeneous quantum Bianchi I model that (a) is covariant with
respect to the actions of such residual diffeomorphisms, and (b) intertwines
both the (signed) volume operator and at least one directional Hubble rate.
That embedding also intertwines all other operators of interest in the
respective loop quantum cosmological models, including their Hamiltonian
constraints. It thus establishes a precise equivalence between dynamics in the
isotropic sector of the Bianchi I model and the quantized isotropic model, and
not just their kinematics. We also discuss the adjoint relationship between the
embedding map defined here and a projection map previously defined by Ashtekar
and Wilson-Ewing. Finally, we highlight certain features that simplify this
reduced embedding problem, but which may not have direct analogues in the
embedding of homogeneous and isotropic loop quantum cosmology into the full
theory of general relativity.
