  We discuss the quantization of vacuum Bianchi I spacetimes in the modified
formalism of loop quantum cosmology recently proposed by Dapor and Liegener.
This modification is based on a regularization procedure where both the
Euclidean and Lorentzian terms of the Hamiltonian are treated independently.
Whereas the Euclidean part has already been dealt with in the literature for
Bianchi I spacetimes, the Lorentzian one has not yet been represented quantum
mechanically. After a brief review of the quantum kinematics and the
quantization of the Euclidean sector, we represent the Lorentzian part of the
Hamiltonian constraint by an operator according to the factor ordering rules of
the Martin-Benito--Mena Marugan--Olmedo prescription. We study the general
properties of this quantum operator and the superselection rules derived
therefrom, resulting in an action similar to that of the Euclidean operator
except that the orientation of the densitized triad is not preserved, a fact
which leads to a generic enlargement of the superselection sectors. We conclude
with an explanation of the mechanism that prevents this enlargement in the
isotropic case and a comment on the effect of alternative prescriptions for the
implementation of the improved dynamics.
