  We shall present here a general apt technique to induce connections along
bundle reductions which is different from the standard restriction. This
clarifies and generalizes the standard procedure to define Barbero-Immirzi (BI)
connection, though on spacetime. The standard spacial BI connection used in LQG
is then obtained by its spacetime version by standard restriction. The general
prescription to define such a reduced connection is interesting from a
mathematical viewpoint and it allows a general and direct control on
transformation laws of the induced object. Moreover, unlike what happens by
using standard restriction, we shall show that once a bundle reduction is
given, then any connection induces a reduced connection with no constraint on
the original holonomy as it happens when connections are simply restricted.
