Does the algebraic structure of a Lie group restrict the possible dimensions of other Lie groups isomorphic to it?

Question

In a recent question, I initially doubted that $\mathbb{C}^\times\cong S^1$, my intuition being that $\mathbb{C}^\times$ has one more "dimension" than $S^1$ - in rigorous terms, $S^1$ is (or rather, can be given the structure of) a 1-dimensional Lie group, and $\mathbb{C}^\times$ is (can be given the structure of) a 2-dimensional Lie group.

My question is rather naive: Given a Lie group $G$ of dimension $n\in\mathbb{N}\cup\{\infty\}$, for which $m\in\mathbb{N}\cup\{\infty\}$ does there exist a Lie group $H$ of dimension $m$ such that $G\cong H$ as groups (forgetting about the manifold structure)? Surely there is at least one $G$ such that the answer isn't all $m$?

Answer 1

I believe all 1-dimensional (connected) real lie groups are abelian, so given a non-abelian lie group there exists no lie group of dimension 1 isomorphic to it.

Answer 2

A related question was asked at MathOverflow. The author argues that there is a unique Lie group structure on $SU(2)$.