@DietrichBurde that article seems a good reference, as far as I understand it deals with nilpotent Lie algebras (which admits compact quotients). Are you aware of something similar on non-nilpotent solvable Lie algebras? In any case, this is a good answer to my question, so I will accept it if you answer. Thank you.

@DietrichBurde True, but nobody answer yet. I will delete those post, keeping this one