  We employ the techniques introduced in the companion papers to derive a
connection formulation of Lorentzian General Relativity coupled to Dirac
fermions in dimensions D+1 > 2 with compact gauge group. The technique that
accomplishes that is similar to the one that has been introduced in 3+1
dimensions already: First one performs a canonical analysis of Lorentzian
General Relativity using the time gauge and then introduces an extension of the
phase space analogous to the one employed in the first paper of this series to
obtain a connection theory with SO(D+1) as the internal gauge group subject to
additional constraints. The success of this method rests heavily on the strong
similarity of the Lorentzian and Euclidean Clifford algebras. A quantisation of
the Hamiltonian constraint is provided.
