  The implications of the SU(2) gauge fixing associated with the choice of
invariant triads in Loop Quantum Cosmology are discussed for a Bianchi I model.
In particular, via the analysis of Dirac brackets, it is outlined how the
holonomy-flux algebra coincides with the one of Loop Quantum Gravity if paths
are parallel to fiducial vectors only. This way the quantization procedure for
the Bianchi I model is performed by applying the techniques developed in Loop
Quantum Gravity but restricting the admissible paths. Furthermore, the local
character retained by the reduced variables provides a relic diffeomorphisms
constraint, whose imposition implies homogeneity on a quantum level. The
resulting picture for the fundamental spatial manifold is that of a cubical
knot with attached SU(2) irreducible representations. The discretization of
geometric operators is outlined and a new perspective for the super-Hamiltonian
regularization in Loop Quantum Cosmology is proposed.
