Definability in the group of infinitesimals of a compact lie group

We show that for G a simple compact Lie group, the infinitesimal subgroup G⁰⁰ is bi-interpretable with a real closed convexly valued field. We deduce that for G an infinite definably compact group definable in an o-minimal expansion of a field, G⁰⁰ is bi-interpretable with the disjoint union of a (possibly trivial) Q-vector space and finitely many (possibly zero) real closed valued fields. We also describe the isomorphisms between such infinitesimal subgroups, and along the way prove that every definable field in a real closed convexly valued field R is definably isomorphic to R.