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
PSL2β 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
PSL2β(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