Corollary 1

Let $\lambda \in \mathfrak{h}^*$ and $\alpha \in B$ be such that $\lambda (H_{\alpha}) + 1\in \mathbf{N}$. Then $Z(-\alpha +s_{\alpha}\lambda )$ is isomorphic to a $\mathfrak{g}$-submodule of $Z(\lambda )$.

Put $m=\lambda (H_{\alpha})$. Let $x= X^{m+1}_{-\alpha}.e\in Z(\lambda )$, and let V be the submodule of $Z(\lambda )$ generated by $x$. Then $x\not= 0$ (Prop. 6). On the other hand, $x\in Z(\lambda )^{\lambda-(m+1)\alpha}$. For $\beta \in B$ and $\beta \not=\alpha , [\mathfrak{g}^{-\alpha},\mathfrak{g}^{\beta}] = 0$ and $\mathfrak{g}^{\beta}.e= 0$, so $\mathfrak{g}^{\beta}.x= 0$. Finally, since $[X_{\alpha},X_{-\alpha}] =-H_{\alpha}$, we have

(§1, no. 1, Lemma 1 (ii)), so

Thus, $x$ is primitive of weight $\lambda -(m+ 1)\alpha$. In view of Prop. 6, the $\mathfrak{g}$-module V is isomorphic to $Z(-\alpha +\lambda -m\alpha ) = Z(-\alpha +s_{\alpha}\lambda )$.