### B.1 Homogeneous models

In Bianchi models the transitive symmetry group acts freely on , which implies that can locally
be identified with the group manifold . The three generators of will be denoted as ,
, with relations , where are the structure constants of
fulfilling for class A models by definition. The Maurer–Cartan form on is given by
with left-invariant 1-forms on , which fulfill the Maurer–Cartan equations
Due to , all homomorphisms are given by , and we can use the embedding
. An invariant connection then takes the form with
matrices generating . The scalar field is given by already
in its final form, because condition (84) is empty for a trivial isotropy group.
Using left-invariant vector fields obeying and with Lie brackets
the momenta canonically conjugate to can be written as with being
canonically conjugate to . Here, is the determinant of the left-invariant metric
on , which is used to provide the density weight of . The symplectic structure
can be derived from

to obtain

with the volume of measured with the invariant metric .
It is convenient to absorb the coordinate volume into the fields by redefining and
. This makes the symplectic structure independent of in accordance with background
independence. These redefined variables automatically appear in holonomies and fluxes through coordinate
integrations.