  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 $W(\mathfrak g)$, the
corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell
$Y_{w'}$ iff $w\prec w'$ 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.
