The Full Symmetric Toda Flow and Intersections of Bruhat Cells

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 $w$, $w^{\mathrm{\prime }}$ in the Weyl group $W(\mathfrak{g})$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w^{\mathrm{\prime }}}$ iff $w\prec w^{\mathrm{\prime }}$ in the Bruhat order on $W(\mathfrak{g})$. Here $\mathfrak{g}$ is a normal real form of a semisimple complex Lie algebra ${\mathfrak{g}}_{\mathbb{C}}$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations.