Yu.B. Chernyakov, G.I. Sharygin, A.S. Sorin and. D.V. Talalaev, "The Full Symmetric Toda Flow and Intersections of Bruhat Cells," SIGMA 16, 115, 8pp. (2020)
Abstract: In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements w, w′ in the Weyl group W(g), the corresponding real Bruhat cell Xw intersects with the dual Bruhat cell Yw′ iff w ≺ w′ in the Bruhat order on W(g). Here g is a normal real form of a semisimple complex Lie algebra gC. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.