Consider the complete flag variety $X$ of a complex semisimple algebraic group $G$. We show that the structure coefficients of the Belkale-Kumar product $\odot_0$, on the cohomology $\mathrm{H}^{*}(X,\mathbf{Z})$, are all either $0$ or $1$. We also derive some consequences. The proof that is mainly geometric also uses new combinatorial results on root systems. Moreover, it is uniform and avoids case by case considerations.