  In this paper we consider a class of hypoelliptic second-order partial
differential operators $\mathcal{L}$ in divergence form on $\mathbb{R}^N$,
arising from CR geometry and Lie group theory, and we prove the Strong and Weak
Maximum Principles and the Harnack Inequality for $\mathcal{L}$. The involved
operators are not assumed to belong to the H\"ormander hypoellipticity class,
nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on the
degeneracy of the second order part; indeed our results hold true in the
infinitely-degenerate case and for operators which are not necessarily sums of
squares. We use a Control Theory result on hypoellipticity in order to recover
a meaningful geometric information on connectivity and maxima propagation, yet
in the absence of any H\"ormander condition. For operators $\mathcal{L}$ with
$C^\omega$ coefficients, this control-theoretic result will also imply a Unique
Continuation property for the $\mathcal{L}$-harmonic functions. The (Strong)
Harnack Inequality is obtained via the Weak Harnack Inequality by means of a
Potential Theory argument, and by a crucial use of the Strong Maximum Principle
and the solvability of the Dirichlet problem for $\mathcal{L}$ on a basis of
the Euclidean topology.
