  In this article, the third of three, we analyse how the Weyl quantisation for
compact Lie groups presented in the second article of this series fits with the
projective-phase space structure of loop quantum gravity-type models. Thus, the
proposed Weyl quantisation may serve as the main mathematical tool to implement
the program of space adiabatic perturbation theory in such models. As we
already argued in our first article, space adiabatic perturbation theory offers
an ideal framework to overcome the obstacles that hinder the direct
implementation of the conventional Born-Oppenheimer approach in the canonical
formulation of loop quantum gravity.
