Corollary 1

Let $(\varpi_{\alpha})_{\alpha\in B}$ be the family of fundamental weights relative to B. Let $\lambda =\sum_{\alpha\in B}m_{\alpha}\varpi_{\alpha}\in P_{++}$. For all $\alpha \in B$, let $E_{\alpha}$ be a simple $\mathfrak{g}$-module of highest weight $\varpi_{\alpha}$. In the $\mathfrak{g}$-module $\bigotimes_{\alpha\in B}(\bigotimes^{m_{\alpha}}E_{\alpha})$, the isotypical component of highest weight $\lambda$ is of length 1.

This follows from Prop. 9 by induction on $\sum_{\alpha\in B}m_{\alpha}$.