  We approach the quasi-isometric classification questions on Lie groups by
considering low dimensional cases and isometries alongside quasi-isometries.
First, we present some new results related to quasi-isometries between Heintze
groups. Then we will see how these results together with the existing tools
related to isometries can be applied to groups of dimension 4 and 5 in
particular. Thus we take steps towards determining all the equivalence classes
of groups up to isometry and quasi-isometry. We completely solve the
classification up to isometry for simply connected solvable groups in dimension
4, and for the subclass of groups of polynomial growth in dimension 5.
