Abstract
We study proper Moufang sets of finite Morley rank for which either the root groups are abelian or the roots groups have no involutions and the Hua subgroup is nilpotent. We give conditions ensuring that the little projective group of such a Moufang set is isomorphic to PSL2(F) for F an algebraically closed field. In particular, we show that any infinite quasisimple L*-group of finite Morley rank of odd type for which (B,N,U) is a split BN-pair of Tits rank 1 is isomorphic to SL2(F ) or PSL2(F) provided that U is abelian. Additionally, we show that same conclusion can reached by replacing the hypothesis that U be abelian with the hypotheses that the intersection of B ∩ N is nilpotent and U is definable and without involutions. As such, we make progress on the open problem of determining the simple groups of finite Morley rank with a split BN-pair of Tits rank 1, a problem tied to the current attempt to classify all simple groups of finite Morley rank.