We prove lower bounds for the fundamental solutions of the non-divergence form operators
\[
∑
i
,
j
a
i
,
j
(
x
,
t
)
X
i
X
j
−
∂
t
and
∑
i
,
j
a
i
,
j
(
x
)
X
i
X
j
,
{\textstyle \sum _{i,j}} a_{i,j}(x,t)\,X_iX_j-\partial _t \quad \text {and}\quad {\textstyle \sum _{i,j}}a_{i,j}(x)\,X_iX_j,
\]
where the
X
i
X_i
’s are Hörmander vector fields generating a stratified group
G
\mathbb {G}
and
(
a
i
,
j
)
i
,
j
(a_{i,j})_{i,j}
is a positive-definite matrix with Hölder continuous entries. We then prove an invariant Harnack inequality for such operators. As a byproduct we also study some relevant properties of the Green functions on bounded domains.