TITLE: Can we promote to a Lie Group Isomorphism?
QUESTION [10 upvotes]: We regard an isomorphism of Lie groups to mean a group isomorphism which is simultaneously a diffeomorphism of the underlying smooth manifold. I'm wondering about how much rigidity is imposed by this definition.
Question: If we have maps $f, g: G\rightarrow H$, where $G, H$ are Lie groups, $f$ is an abstract group isomorphism, and $g$ is a diffeomorphism, must $G$ and $H$ be isomorphic as Lie groups? 
I think there should be a (possibly easy) counterexample, but neither I nor the professors I've asked could immediately find one. 
EDIT: assume $G$ and $H$ are connected.

REPLY [3 votes]: All you need is love:-)) Not really, just take a family of complex nilpotent Lie algebras $L_a$ depending on a complex parameter $a$. Such a family exists in dimension 7, or maybe even 6 (I do not remember the nilpotent classification).
Now consider a wild automorphism of complex numbers $\phi$. Then $L_a$ and $L_{\phi (a)}$ for a generic $a$ are isomorphic as Lie algebras over ${\mathbb Q}$ but not over ${\mathbb R}$ or ${\mathbb C}$. Love is all you need((-: the corresponding simply connected Lie groups are isomorphic as abstract groups but not as Lie groups.