Rough similarity of left-invariant Riemannian metrics on some Lie groups

We consider Lie groups that are either Heintze groups or Sol-type groups, which generalize the three-dimensional Lie group SOL. We prove that all left-invariant Riemannian metrics on each such a Lie group are roughly similar via the identity map. This allows us to reformulate in a common framework former results by Le Donne–Xie, Eskin–Fisher–Whyte, Carrasco Piaggio, and recent results of Ferragut and Kleiner–Müller–Xie, on quasi-isometries of these solvable groups.