Definability in the group of infinitesimals of a compact Lie group

Abstract: We show that for $G$ a simple compact Lie group, the infinitesimal subgroup $G^{00}$ is bi-intepretable with a real closed valued field. We deduce that for $G$ an infinite definably compact group definable in an o-minimal expansion of a field, $G^{00}$ is bi-interpretable with the disjoint union of a (possibly trivial) $\mathbb{Q}$-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 {\em definable} field in a real closed convexly valued field $R$ is definably isomorphic to $R$.