Is it possible to describe, up to isomorphism, all Lie groups $G$ whose underlying manifold is diffeomorphic to $\mathbb{R}^n$ (with its standard smooth structure)?


In fact, I haven't found any such group yet, other than the standard $(\mathbb{R}^n, +)$.



Classification of Lie group structures on $\mathbb{R}^n$
YCor claims [here](https://math.stackexchange.com/questions/475385/under-what-conditions-is-the-exponential-map-on-a-lie-algebra-injective/1592257#1592257) that the contractible Lie groups (this is equivalent to being diffeomorphic to $\mathbb{R}^n$, since a connected Lie group is diffeomorphic to the product of a Euclidean space times its maximal compact) are exactly the Lie groups of the form $S \rtimes \widetilde{SL\_2(\mathbb{R})}^k$ where $S$ is a simply connected solvable Lie group and $\widetilde{SL\_2(\mathbb{R})}$ is the universal cover of $SL\_2(\mathbb{R})$.


So the classification reduces to the classification of solvable Lie algebras (then of semidirect products of these Lie algebras with $\mathfrak{sl}\_2(\mathbb{R})^k$), which I understand to be hopeless.


