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Invariant Riemannian metrics and f-structures on flag manifolds

It turns out that if a reductive complement m of a homogeneous reductive space G/H possesses a number of properties (including its decomposability into an orthogonal sum of three Ad(H)-invariant irreducible subspaces) then there exists a simple way of determining whether an f -structure F on this space belongs to the classes G1f , NKf and Kill f of generalized Hermitian geometry with respect to any invariant Riemannian metric g (provided that F is a metric f -structure with respect to g). Two examples (of flag manifolds) are included.