Alekseevski proves for Heintze groups (a special class of solvable Lie groups) that any such group admits a (left-invariant) metric which is isometric to a purely real Heintze group (again equipped with some left-invariant metric).
Can the same be said for arbitrary solvable Lie groups? Here by purely real, I mean that the adjoint action has purely real eigenvalues when acting as an automorphism of the Lie algebra.
In case it is relevant, I am only interested in those solvable Lie groups which admit nonpositively curved (CAT(0)) metrics.
