  Consider the complete flag variety $X$ of any complex semi-simple algebraic
group $G$. We show that the structure coefficients of the Belkale-Kumar product
$\odot_0$, on the cohomology ${\operatorname H}^{*}(X,\mathbb{Z})$, are all
either $0$ or $1$. We also derive some consequences. The proof contains a
geometric part and uses a combinatorial result on root systems. The geometric
method is uniform whereas the combinatorial one is proved by reduction to small
ranks and then, by direct checkings.
