We count the number of connected components in the space of triples of transverse flags in any flag manifold of SO_0 (p, q). We compute the effect the involution of the unipotent radical has on those components and deduce that for certain parabolic subgroups P_Θ , any P_Θ -Anosov subgroup is virtually isomorphic to either a surface group of a free group. We give examples of Anosov subgroups which are neither free nor surface groups for some sets of roots which do not fall under the previous results. As a consequence of the methods developed here, we get an explicit algorithm based on computation of minors to check if a unipotent matrix in SO_0 (p, q) belong to the Θ-positive semigroup U_0 Θ when p ̸ = q.