  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
$\operatorname{PSL}_2$ 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
$\operatorname{PSL}_2(F)$ 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.
