Motives, cohomological invariants and Freudenthal magic square

Abstract

We investigate cohomological invariants and motivic invariants of semisimple algebraic groups arising in the Freudenthal magic square. Besides, we show that if the Rost invariant of a strongly inner group of type $E_7$ is a sum of at most two symbols modulo $2$, then it is isotropic over an odd degree field extension, and use this fact to give a different proof of a result of Petrov and Rigby. Moreover, we give a motivic interpretation of a result of Garibaldi and Petersson about a cohomological invariant of degree $5$ for certain groups of type $^2E_6$ which detects their isotropy.