  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 H\"ormander 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 H\"ormander condition is satisfied.
