Definability in the infinitesimal subgroup of a simple compact Lie group

Joint work with Kobi Peterzil.Let G be a simple compact Lie group, for example G=SO_3(R). We consider the structure of definable sets in the subgroup G^{00} of infinitesimal elements. In an aleph_0-saturated elementary extension of the real field, G^{00} is the inverse image of the identity under the standard part map, so is definable in the corresponding valued field. We show that the pure group structure on G^{00} recovers the valued field, making this a bi-interpretation. Hence the definable sets in the group are as rich as possible.