Corollary

Let $\mathscr{X}$ be a finite R-saturated subset of P. There exists a finite dimensional $\mathfrak{g}$-module whose set of weights is $\mathscr{X}$.

Since $\mathscr{X}$ is stable under $W,\mathscr{X}$ is the smallest R-saturated set containing $\mathscr{X}\cap P_{++}$. By Prop. 5 (i), $\mathscr{X}$ is the set of weights of $\bigoplus_{\lambda\in\mathscr{X}\cap P_{++}}E(\lambda )$.