Does there exist (as elementary as possible) a generalized flag manifold (e.g. sphere, projective space, Grassmanian, etc.) of a simple classical Lie group (SL, SO or Sp over real or complex field), such that $F_4$ is acting transitively on it with stabilizer given by parabolic subgroup? In other words, is there a Lie group G (equal to SL, SO or Sp) and a parabolic subgroup $P< G$, such that G/P is equal to $F_4/Q$, where Q is parabolic subgroup of $F_4$ given by intersection of $F_4$ with P?
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