Rational subgroups and invariants of F4

The cyclicity problem for Albert algebras

“…By the theorem, therefore, we may assume that A contains a cyclic cubic subfield. Applying ( [4], Cor. 3.2), we conclude that the automorphism group scheme of A contains a subgroup of type 3 D 4 defined over k. But the automorphism group scheme of A is contained in its structure group scheme.…”