We classify the groups from the title. The result is interpreted in terms of nearfields, and applied to a problem in the model theory of permutation groups.

Let G be a group definable in an o-minimal structure M. In this paper we show: Theorem. If G is a two-dimensional definably connected nonabelian group, then G is centerless and G is isomorphic to R+⋊R∗>0, for some real closed field R. Theorem. If G is a three-dimensional nonsolvable, centerless, definably connected group, then either G ≅ SO3(R) or G ≅ PSL2(R), for some real closed field R.

We prove the following theorem: Let G be a connected centerless non-solvable group of Morley rank 3 whose proper definable subgroups have rank ⩽1. Then G has no involutions and for all x ϵ G, CG(x) is connected. Our method permits us to deduce that SOn(R) is not stable for n ⩾ 3.

This shows that D _ B. Similarly C __C_ A. Changing the roles of A and C, B and D we get the inverse inclusion.The proof of the theorem is now complete.