## A rank-three condition for invariant (1,2)-symplectic almost Hermitian structures on flag manifolds |

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1,2)-symplectic structures 2fey 4-player abelian ideal affine iacs affine Weyl group bijection Borel subalgebra C. J. C. Negreiros Cartan-Killing form complex Lie complex structures concluding the proof condition of Lemma cone cone-free condition cone-free iacs cone-free property defined Definition 3.1 denote doubly-laced root system Dynkin diagram extracted from q extracted triples Fernando Torres form of G form the coordinates Hence Hermitian structures highest root Homogeneous spaces induction inequalities invariant almost complex Invariant almost Hermitian invariant structures Lemma Lie algebras Lie group long roots Math matrix maximal flag manifold maximal torus n-player tournament Proposition 8.2 prove quadruples rank-two restriction Riemannian metric Rojas-Medar Santos satisfies the cone-free semi-simple Lie set of integers set of short short roots simple roots simply-laced root system structure on F structures on flag subalgebra subspace spanned system of roots tangent space Theorem 7.1 triples extracted zero