  A simple Almost-Riemannian Structure on a Lie group G is defined by a linear
vector field (that is an infinitesimal automorphism) and dim(G) -- 1
left-invariant ones. It is first proven that two different ARSs are isometric
if and only if there exists an isometry between them that fixes the identity.
Such an isometry preserves the left-invariant distribution and the linear
field. If the Lie group is nilpotent it is an automorphism. These results are
used to state a complete classification of the ARSs on the 2D affine and the
Heisenberg groups.
