All you need is love:-)) Not really, just take a semidirect product family of complex nilpotent Lie algebras $G_{a,b}={\mathbb R}\ltimes {\mathbb R}^2$L_a$depending on a complex parameter$a$. Such a family exists in dimension 7, or maybe even 6 (I do not remember the nilpotent classification).The action is diagonal with two eigenvalues Now consider a wild automorphism of complex numbers$a$\phi$. Then $L_a$ and $b$. Its L_{\phi (a)}$for a generic$a$are isomorphic as Lie group isomorphism class is determined by algebras over$a/b$. Not let {\mathbb Q}$ but not over $\phi : {\mathbb R}\rightarrow {\mathbb {\mathbb R}$ be an isomorphism of or ${\mathbb Q}$-vector spacesC}$. Then$G_{a,b}$Love is obviously all you need((-: the corresponding simply connected Lie groups are isomorphic to$G_{\phi(a),\phi(b)}$as abstract groups but$\phi(a)/\phi(b)$could be easily different from$a/b\$.not as Lie groups.