Quantization of Lorentzian 3D gravity by partial gauge fixing

D = 2 + 1 gravity with a cosmological constant Λ has been shown by Bonzom and Livine to present a Barbero–Immirzi-like ambiguity depending on a parameter γ. We make use of this fact to show that, for Λ >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.