to the nilradical of a Borel subalgebra of the semisimple Lie algebra ${\widetilde}{\mathfrak{g}}_e$ with the root system ${\widetilde}\Phi\times A_1$, and the Kostant cascade of ${\widetilde}{\mathfrak{n}}_e$ is ${\widetilde}{\mathcal{B}}\cup\{\beta_2\}$. Denote by ${\widetilde}{\mathfrak{h}}_e$ the corresponding Cartan subalgebra of ${\widetilde}{\mathfrak{g}}_e$. By [@Kostant2 Theorem 6] (see also [@Joseph1 Lemma 4.4]), $Z({\mathfrak{n}})$ (respectively, $Z({\widetilde}{\mathfrak{n}}_e)$) is a direct sum of 1-dimensional weight spaces of ${\mathfrak{h}}$ (respectively, of ${\widetilde}{\mathfrak{h}}_e$) with respect to the adjoint action of the corresponding Cartan subalgebras. Since $[h,e_{\beta_1}]=0$ for all $h\in{\widetilde}{\mathfrak{h}}_e$, the algebra ${\widetilde}{\mathfrak{h}}_e$ naturally acts on ${\mathcal{A}}_s$, and so on $U({\widetilde}{\mathfrak{n}}_e)\otimes{\mathcal{A}}_s$. We define the result of this action by $h.x$, $h\in{\widetilde}{\mathfrak{h}}_e$, $x\in U({\widetilde}{\mathfrak{n}}_e)\otimes{\mathcal{A}}_s$. Hence, according to (\[formula:weights\_of\_xi\_i\]), it is enough to check that $r({P}_{\beta})$ is a nonzero ${\widetilde}{\mathfrak{h}}_e$-weight element of weight ${\widetilde}{\varpi}_{\beta}$, where $${\widetilde}{\varpi}_{\beta}=\begin{cases}
{\epsilon}_3+\ldots+{\epsilon}_{2k}&\text{for }\beta={\epsilon}_{2k-1}+{\epsilon}_{2k},~k\geq2,\\
{\epsilon}_3+\ldots+{\epsilon}_n&\text{for }\beta={\epsilon}_n,~\Phi=B_n,~n\text{ odd},\\
{\epsilon}_3+\ldots+{\epsilon}_{n-1}-{\epsilon}_n&\text{for }\beta={\epsilon}_{n-1}-{\epsilon}_n,~\Phi=D_n,~n\text{ even},\\
2{\epsilon}_3+\ldots+2{\epsilon}_{2k-1}&\text{for }\beta={\epsilon}_{2k-1}-{\epsilon}_{2k},~k\geq2,~2k<n\text{ if }\Phi=D_n.\\
\end{cases}$$

To prove that $r({P}_{\beta})$ is an ${\widetilde}{\mathfrak{h}}_e$-weight element of weight ${\widetilde}{\varpi}_{\beta}$, denote the result of the natural (adjoint) action of ${\widetilde}{\mathfrak{h}}_e$ on $U({\mathfrak{n}})$ by $h\cdot x$, $h\in{\widetilde}{\mathfrak{h}}_e$, $x\in U({\mathfrak{n}})$. As above, since ${\widetilde}{\mathfrak{h}}_e\cdot e_{\beta_1}=0$, the algebra ${\widetilde}{\mathfrak{h}}_e$ naturally acts on $U({\mathfrak{n}})/J_0$ by the formula $h\cdot r(x)=r(h\cdot x)$. We claim that this action coincides with the action of ${\widetilde}{\mathfrak{h}}_e$ on $U({\widetilde}{\mathfrak{n}}_e)\otimes{\mathcal{A}}_s$ defined above, i.e., that $h.x=h\cdot x$ 