Simple algebraic and semialgebraic groups over real closed fields

We continue the investigation of infinite, definably simple groups which ar e definable in o-minimal structures. In Definably simple groups in o-minima l structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic lan guage, is that every such group is either bi-interpretable with an algebrai cally closed field of characteristic zero (when the group is stable) or wit h a real closed field (when the group is unstable). It follows that every a bstract isomorphism between two unstable groups as above is a composition o f a semialgebraic map with a field isomorphism. We discuss connections to t heorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.