The geometry of involutions in ranked groups with a ti-subgroup

We revisit the geometry of involutions in groups of finite Morley rank. The focus is on specific configurations where, as in PGL2(K), the group has a subgroup whose conjugates generically cover the group and intersect trivially. Our main result is the subtle yet strong statement that in such configurations the conjugates of the subgroup may not cover all strongly real elements. As an application, we unify and generalise numerous results, both old and recent, which have exploited a similar method; though in fact we prove much more. We also conjecture that this path leads to a new identification theorem for PGL2(K), possibly beyond the finite Morley rank context.