In this paper we investigate the well-posedness of the Cauchy problem for the wave equation for sums
of squares of vector fields on compact Lie groups. We obtain the loss of regularity for solutions to the
Cauchy problem in local Sobolev spaces depending on the order to which the Hörmander condition is
satisfied, but no loss in globally defined spaces. We also establish Gevrey well-posedness for equations
with irregular coefficients and/or multiple characteristics. As in the Sobolev spaces, if formulated in local
coordinates, we observe well-posedness with the loss of local Gevrey order depending on the order to which
the Hörmander condition is satisfied