  We shall show that no reductive splitting of the spin group exists in
dimension 3 \leq m \leq 20 other than in dimension m = 4. In dimension 4 there
are reductive splittings in any signature. Euclidean and Lorentzian signatures
are reviewed in particular and signature (2, 2) is investigated explicitly in
detail. Reductive splittings allow to define a global SU(2)-connection over
spacetime which encodes in an weird way the holonomy of the standard spin
connection. The standard Barbero-Immirzi (BI) connection used in LQG is then
obtained by restriction to a spacelike slice. This mechanism provides a good
control on globality and covariance of BI connection showing that in dimension
other than 4 one needs to provide some other mechanism to define the analogous
of BI connection and control its globality.
