  Let $G$ be an affine group over a field of characteristic not two. A
$G$-torsor is called isotropic if it admits reduction of structure to a proper
parabolic subgroup of $G$. This definition generalizes isotropy of affine
groups and involutions of central simple algebras. When does $G$ admit
anisotropic torsors? Building on work of J. Tits, we answer this question for
simple groups. We also give an answer for connected and semisimple $G$ under
certain restrictions on its root system.
