system ${\widetilde}\Phi=B_{n-2}$ (respectively, ${\widetilde}\Phi=D_{n-2}$), and ${\mathfrak{k}}$ is isomorphic to the $(4n-5)$-dimensional (respectively, $(4n-7)$-dimensional) Heisenberg algebra. Indeed, $[e_{{\epsilon}_1-{\epsilon}_i},e_{{\epsilon}_2+{\epsilon}_i}]=e_{\beta_1}$ for $3\leq i\leq n$, $[e_{{\epsilon}_1},e_{{\epsilon}_2}]=2e_{\beta_1}$, and $[e_{\alpha},e_{\gamma}]=0$ for all other $\alpha,~\gamma\in({\mathcal{R}}_1\cup{\mathcal{R}}_2)\setminus\{\beta_2\}$.

Given $c_1\in{\mathbb{C}}^{\times}$, denote by $J_1$ the ideal of $U({\mathfrak{k}})$ generated by $e_{\beta_1}-c_1$, then, clearly, $U({\mathfrak{k}})/J_1\cong{\mathcal{A}}_s$, where $s=2n-3$ for $B_n$ and $s=2n-4$ for $D_n$. Denote ${\widetilde}{\mathfrak{n}}_e={\widetilde}{\mathfrak{n}}\oplus{\mathbb{C}}e$. Note that ${\mathfrak{k}}$ is an ideal of the Lie algebra ${\mathfrak{n}}$, so, given $x\in{\widetilde}{\mathfrak{n}}_e$, one can consider ${\mathrm{ad}_{x}}$ as a derivation of ${\mathfrak{k}}$. Since $e_{\beta_1}-c_1$ is a central element of $U({\mathfrak{n}})$, one has ${\mathrm{ad}_{x}}(J_1)\subseteq J_1$, so ${\mathrm{ad}_{x}}$ can be considered as a derivation of ${\mathcal{A}}_s$. It is well known (see, e.g., [@Dixmier3 Lemma 10.1.2, Lemma 10.1.3]) that there exist the unique element $\theta(x)\in{\mathcal{A}}_s$ such that ${\mathrm{ad}_{x}}(y)=[\theta(x),y]$ for all $y\in{\mathcal{A}}_s$, and $\theta\colon{\widetilde}{\mathfrak{n}}_e\to{\mathcal{A}}_s$ is a morphism of Lie algebras. Furthermore, there exists the unique epimorphism of associative algebras $$r\colon U({\mathfrak{n}}){\twoheadrightarrow}U({\widetilde}{\mathfrak{n}}_e)\otimes{\mathcal{A}}_s$$ such that $r(y)=1\otimes \overline y$ for $y\in{\mathfrak{k}}$ and $r(x)=x\otimes1+1\otimes\theta(x)$ for $x\in{\widetilde}{\mathfrak{n}}_e$. (Here $\overline a$ is the image of an element $a\in U({\mathfrak{k}})$ under the canonical projection $U({\mathfrak{k}}){\twoheadrightarrow}U({\mathfrak{k}})/J_1\cong{\mathcal{A}}_s$.) It turns out that the kernel of $r$ coincides with the ideal $J_0$ of $U({\mathfrak{n}})$ generated by $e_{\beta_1}-c_1$ [@Dixmier3 Lemma 10.1.5]. One can easily check by direct computation that $r(P_{\beta_2})=c_1e\otimes1$. We are 