  We give a simplified proof of Tits' classification of semisimple algebraic
groups that remains valid over semilocal rings. In particular, we provide
explicit necessary and sufficient conditions that anisotropic groups of a given
type appear as anisotropic kernels of semisimple groups of a given Tits index.
We also give a new proof of the existence of all indices of exceptional inner
type using the notion of canonical dimension of projective homogeneous
varieties.
