  We consider a finite system $\{X_1, X_2, \ldots, X_n\}$ of complete vector
fields acting on smooth manifolds $M$ equipped with a smooth positive measure.
We assume that the system satisfies H\"ormander's condition and generates a
finite dimensional Lie algebra of type (R). We investigate the sum of squares
of the vector fields operator corresponding to this system which can be viewed
as a generalisation of the notion of Grushin operators. In this setting we
prove the Poincar\'e inequality and Li-Yau estimates for the corresponding heat
kernel as well as the doubling condition for the optimal control metrics
defined by the system. We discuss a surprisingly broad class of examples of
described setting.
