In this paper we determine isomorphism classes of connected solvable Lie groups with some conditions such that their group -algebras have stable rank one, and give its applications. Also, we show that stable rank of group -algebras of connected Lie groups of type I is estimated in terms of their closed normal subgroups and quotient groups.
"Dimension theory of group -algebras of connected Lie groups of type I." J. Math. Soc. Japan 52 (3) 583 - 590, July, 2000. https://doi.org/10.2969/jmsj/05230583