  Lying at the intersection of Ado's theorem and the Nash embedding theorem, we
consider the problem of finding faithful representations of Lie groups which
are simultaneously isometric embeddings. Such special maps are found for a
certain class of solvable Lie groups which includes all Einstein and Ricci
soliton solvmanifolds, as well as all Riemannian 2-step nilpotent Lie groups.
As a consequence, we extend work of Tamaru by showing that all Einstein
solvmanifolds can be realized as submanifolds (in the submanifold geometry) of
a symmetric space.
