Moufang sets of finite Morley rank of odd type

We show that for a wide class of groups of finite Morley rank the presence of a split BN  -pair of Tits rank 1 forces the group to be of the form PSL2PSL2 and the BN  -pair to be standard. Our approach is via the theory of Moufang sets. Specifically, we investigate infinite and so-called hereditarily proper Moufang sets of finite Morley rank in the case where the little projective group has no infinite elementary abelian 2-subgroups and show that all such Moufang sets are standard (and thus associated to PSL2(F)PSL2(F) for F an algebraically closed field of characteristic not 2) provided the Hua subgroups are nilpotent. Further, we prove that the same conclusion can be reached whenever the Hua subgroups are L-groups and the root groups are not simple.