  In this article, the second of three, we discuss and develop the basis of a
Weyl quantisation for compact Lie groups aiming at loop quantum gravity-type
models. This 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. Additionally, we conjecture the existence
of a new form of the Segal-Bargmann-Hall "coherent state" transform for compact
Lie groups $G$, which we prove for $G=U(1)^{n}$ and support by numerical
evidence for $G=SU(2)$. The reason for conjoining this conjecture with the main
topic of this article originates in the observation, that the coherent state
transform can be used as a basic building block of a coherent state
quantisation (Berezin quantisation) for compact Lie groups $G$. But, as Weyl
and Berezin quantisation for $\mathbb{R}^{2d}$ are intimately related by heat
kernel evolution, it is natural to ask, whether a similar connection exists for
compact Lie groups, as well. Moreover, since the formulation of space adiabatic
perturbation theory requires a (deformation) quantisation as minimal input, we
analyse the question to what extent the coherent state quantisation, defined by
the Segal-Bargmann-Hall transform, can serve as basis of the former.
