Let be an exceptional simple algebraic group over a field , and let be a projective -homogeneous variety such that splits over . We classify such varieties . This classification allows us to relate the Rost invariant of groups of type and their isotropy and to give a two-line proof of the triviality of the kernel of the Rost invariant for such groups. Apart from this, it plays a crucial role in the solution of a problem posed by Serre for groups of type
Duke Math. J.
152(1):
155-173
(15 March 2010).
DOI: 10.1215/00127094-2010-010
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