On some local-global principles for isotropy and splitting of algebraic groups over global fields

We consider certain local-global principles related with the isotropy and some splitting problems for connected linear algebraic groups over global fields. The main tools are certain reciprocity law for Tits indices of almost simple groups due to Prasad and Rapinchuk, Harder's Hasse principle for homogeneous projective spaces of reductive groups for number fields and their extensions to global function fields.