  For every number field and every Cartan Killing type, there is an associated
split simple algebraic group. We examine whether the corresponding arithmetic
subgroups are profinitely solitary so that the commensurability class of the
profinite completion determines the commensurability class of the group among
finitely generated residually finite groups. Assuming Grothendieck rigidity, we
essentially solve the problem by Galois cohomological means.
