  We introduce a master constraint operator on the kinematical Hilbert space of
loop quantum gravity representing a set of gauge conditions which classically
fix the densitized triad to be diagonal. We argue that the master constraint
approach provides a natural and systematic way of carrying out the quantum
gauge-fixing procedure which underlies the model known as quantum-reduced loop
gravity. The Hilbert space of quantum-reduced loop gravity is obtained as a
particular space of solutions of the gauge-fixing master constraint operator.
We give a concise summary of the fundamental structure of the quantum-reduced
framework, and consider several possible extensions thereof, for which the
master constraint formulation provides a convenient starting point. In
particular, we propose a generalization of the standard Hilbert space of
quantum-reduced loop gravity, which may be relevant in the application of the
quantum-reduced model to physical situations in which the Ashtekar connection
is not diagonal.
