  De Sitter Chern-Simons gravity in D = 1 + 2 spacetime is known to possess an
extension with a Barbero-Immirzi like parameter. We find a partial gauge fixing
which leaves a compact residual gauge group, namely SU(2). The compacticity of
the residual gauge group opens the way to the usual LQG quantization
techniques. We recall the exemple of the LQG quantization of SU(2) CS theory
with cylindrical space topology, which thus provides a complete LQG of a
Lorentzian gravity model in 3-dimensional space-time.
