Quantization of Lorentzian 3D gravity by partial gauge fixing

D = 2 + 1 gravity with a cosmological constant Lambda has been shown by Bonzom and Livine to present a Barbero-Immirzi-like ambiguity depending on a parameter gamma. We make use of this fact to show that, for Lambda > 0, the Lorentzian theory can be partially gauge fixed and reduced to an SU(2) Chern-Simons theory. We then review the already known quantization of the latter in the framework of loop quantization for the case of space being topologically a cylinder. We finally construct, in the same setting, a quantum observable which, although non-trivial at the quantum level, corresponds to a null classical quantity.