for all $h\in{\widetilde}{\mathfrak{h}}_e$, $x\in U({\widetilde}{\mathfrak{n}}_e)\otimes{\mathcal{A}}_s$. Indeed, if $y\in{\mathfrak{k}}$, then $h\cdot r(y)=r([h,y])=1\otimes\overline{[h,y]}=h.(1\otimes y)=h.r(y)$, as required. On the other hand, if $e_{\alpha}\in{\widetilde}{\mathfrak{n}}_e$ for some $\alpha\in\Phi^+$, then, by [@Joseph1 Subsection 4.8], $\theta(e_{\alpha})$ is a linear combination of elements of the form $\overline{e}_{\alpha+\gamma}\overline{e}_{\beta_1-\gamma}$, $\gamma\in({\mathcal{R}}_1\cup{\mathcal{R}}_2)\setminus\{\beta_1,\beta_2\}$. We conclude that $h(1\otimes\theta(e_{\alpha}))=(\beta_1+\alpha)(h)1\otimes\theta(\alpha)=\alpha(h)1\otimes\theta(e_{\alpha})$, because $\beta_1({\widetilde}{\mathfrak{h}}_e)=0$. Thus, $$h.r(e_{\alpha})=[h,e_{\alpha}]\otimes1+1\otimes\alpha(h)\theta(e_{\alpha})=\alpha(h)1\otimes r(e_{\alpha})=h\cdot r(e_{\alpha}).$$ It remains to note that ${P}_{\beta}$ is an ${\mathfrak{h}}$-weight element of $U({\mathfrak{n}})$ of weight $\beta_1+{\widetilde}{\varpi}_{\beta}$, but $\beta_1({\widetilde}{\mathfrak{h}}_e)=0$. To show that $r({P}_{\beta})\neq0$, recall that the kernel of $r$ is $J_0$. If ${P}_{\beta}\in J_0$ (i.e., if ${P}_{\beta}=(e_{\beta_1}-c_{\beta_1})a$ for some $a\in U({\mathfrak{n}})$), then, clearly, $a\in Z({\mathfrak{n}})$. But this contradicts the fact that ${P}_{\beta}$ and $e_{\beta_1}$ are algebraically independent.

So, given $\beta\in{\widetilde}{\mathcal{B}}$, there exists the unique $a_{\beta}\in{\mathbb{C}}^{\times}$ such that $r({P}_{\beta})=a_{\beta}{\widetilde}{P}_{\beta}\otimes1$. Consequently, $r(J_c)$ is generated by $e\otimes1-c_{\beta_1}^{-1}c_{\beta_2}$ and by ${\widetilde}{P}_{\beta}\otimes1-{\widetilde}c_{\beta}$, $\beta\in{\widetilde}{\mathcal{B}}$, where ${\widetilde}c_{\beta}=a_{\beta}^{-1}c_{\beta}$. In particular, ${\widetilde}c_{\beta}\neq0$ if $\beta\in{\widetilde}{\mathcal{B}}$ is not a simple root of ${\widetilde}\Phi^+$. Now we will use the induction on ${\mathrm{rk}\,}\Phi$ to prove that $J_c$ is primitive. One can check the base ${\mathrm{rk}\,}\Phi\leq3$ directly using explicit formulas from Subsection \[sst:center\_fd\]. Denote by ${\widetilde}J_c$ the ideal of $U({\widetilde}{\mathfrak{n}})$ generated by ${\widetilde}{P}_{\beta}-{\widetilde}c_{\beta}$, $\beta\in{\widetilde}{\mathcal{B}}$. By the inductive assumption, ${\widetilde}J_c$ is a primitive ideal of $U({\widetilde}{\mathfrak{n}})$, so $U({\widetilde}{\mathfrak{n}})/{\widetilde}J_c\cong{\mathcal{A}}_t$ for certain $t$. Taking in account that $U({\widetilde}{\mathfrak{n}}_e)=U({\widetilde}{\mathfrak{n}})\otimes{\mathbb{C}}[e]$, and 