  Vielbeins are necessary when coupling General Relativity (GR) to fermionic
matter. This enhances the gauge group of GR to include local Lorentz
transformations.
  In view of a reduced phase space formulation of quantum gravity, in this work
we completely gauge fix that Lorentz gauge symmetry by using a so-called
triangular gauge. Having solved the Gauss constraints already classically opens
access to new Hilbert space representations which are free of the complications
that otherwise arise due to a non Abelian gauge symmetry.
  In that sense, a connection formulation as being pursued in Loop Quantum
Gravity is no longer the only practicable option and other less dimension
dependent representations e.g. based on triads and even metrics suggest
themselves. These formulations make it easier to identify states representing
non-degenerate quantum geometries and thus to investigate the hypersurface
deformation algebra which implicitly assumes non-degeneracy.
