Lie groups definable in o-minimal theories

In this talk we will work out a complete characterization of which Lie groups admit a “definable copy”. This is, characterize for which Lie groups G one can find a group H definable in an o-minimal expansion of the real field, and such that G and H are isomorphic.
When the answer is positive, the definable copy H that we find is definable in the language of exponential ordered fields, and it is such that any Lie automorphism of H is definable.