Given a connected unimodular Lie group G having polynomial volume growth, and left invariant vector fields X 1, …, X k which generate the Lie algebra of G, consider the operator L a= ∑ i,j=1 k X ia ijX j , where a: G→ R k ⊗ R k is a smooth symmetric matrix valued function which satisfies a α ¦ξ¦ 2 ⩽ ∑ i, j = 1 k a ij(x) ξ iξ j ⩽ α −1 ¦ξ¦ 2, x ϵ G, ξϵR k , for some α ϵ ]0, 1 ]. What we show is that the corresponding heat-flow semigroup e tLa admits a kernel which satisfies (two-sided) Gaussian estimates in terms of the control distance determined by the X i 's. Moreover, the estimates can be made to depend on a only through α. We also prove Harnack inequalities and holder regularity of solutions. We end with a discussion of some other sub-elliptic situations.