Assume that $\Phi=A_{n-1}$ and ${\mathrm{col}}(\alpha)<n-i+1$, ${\mathrm{col}}(\gamma)<n-i+1$, or that $\Phi=C_n$ and ${\mathrm{col}}(\alpha)>0$, ${\mathrm{col}}(\gamma)>0$. Then, as it is shown in the proof of [@IgnatyevPenkov1 Lemma 1.5], $$[p_{\alpha},q_{\gamma}]=\begin{cases}
1,&\text{if }\alpha=\gamma,\\
0&\text{otherwise}.\\
\end{cases}$$ Thus, $U({\mathfrak{n}})/J_c\cong{\mathcal{A}}_s$ for $s=|\Phi^+\setminus{\mathcal{B}}|/2$, as required.

Now we will formulate our first main result (cf. Theorem \[theo:ideal\_fd\_An\_Cn\]).

**Theorem ..** [*Suppose $\Phi$ is of type $B_n$ or $D_n$. The following conditions on a primitive ideal $J\subset U({\mathfrak{n}})$ are equivalent $$\begin{split}
&\text{\textup{i)} $J$ is centrally generated \textup{(}or\textup{,} equivalently\textup{,} $J=J_c$\textup{)}};\\
&\text{\textup{ii)} the scalars $c_{\beta}$\textup, $\beta\in{\mathcal{B}}\setminus\Delta$\textup, are nonzero};\\
&\text{\textup{iii)} $J=J(f_{\xi})$ for a Kostant form $f_{\xi}\in{\mathfrak{n}}^*$}.\\
\end{split}$$ If these\[theo:ideal\_fd\_Bn\_Dn\] conditions are satisfied then the map $\xi$ is reconstructed by $J$ $$\xi(\beta)\label{formula_xi_via_c_fd_Bn_Dn}=
\begin{cases}
c_{{\epsilon}_1+{\epsilon}_2}&\text{for }\beta={\epsilon}_1+{\epsilon}_2,\\
c_{{\epsilon}_{2k-1}+{\epsilon}_{2k}}c_{{\epsilon}_{2k-3}+{\epsilon}_{2k-2}}^{-1}&\text{for }\beta={\epsilon}_{2k-1}+{\epsilon}_{2k},~k\geq2,\\
c_{{\epsilon}_1-{\epsilon}_2}c_{{\epsilon}_1+{\epsilon}_2}^{-1}&\text{for }\beta={\epsilon}_1-{\epsilon}_2,\\
c_{{\epsilon}_{2k-1}-{\epsilon}_{2k}}c_{{\epsilon}_{2k-1}+{\epsilon}_{2k}}^{-1}c_{{\epsilon}_{2k-3}+{\epsilon}_{2k-2}}^{-1}&\text{for }\beta={\epsilon}_{2k-1}-{\epsilon}_{2k},~k\geq2,~2k<n\text{ if }\Phi=D_n,\\
c_{{\epsilon}_{n-1}-{\epsilon}_n}c_{{\epsilon}_{n-1}+{\epsilon}_n}^{-1}&\text{for }\beta={\epsilon}_1-{\epsilon}_n,~\Phi=D_n,~n\text{ even},\\
c_{{\epsilon}_n}c_{{\epsilon}_{n-2}+{\epsilon}_{n-1}}^{-1}&\text{for }\beta={\epsilon}_n,~\Phi=B_n,~n\text{ odd}.\\
\end{cases}$$*]{}

Again, as for $A_{n-1}$ and $C_n$, one of the most important steps is to prove that if condition (ii) is satisfied then the centrally generated ideal $J_c$ is primitive. To do this in the orthogonal case, we need some additional technical (but important) facts. Put $\beta_1={\epsilon}_1+{\epsilon}_2$, $\beta_2={\epsilon}_1-{\epsilon}_2$, $e=e_{\beta_2}$, and denote ${\widetilde}{\mathfrak{n}}=\langle e_{\alpha},~\alpha\in\Phi^+$, ${\mathrm{row}}(\alpha)>2\rangle_{{\mathbb{C}}},~{\mathfrak{k}}=\langle e_{\alpha},~{\mathrm{row}}(\alpha)\leq2,~\alpha\neq\beta_2\rangle_{{\mathbb{C}}}$. For $\Phi=B_n$ (respectively, for $\Phi=D_n$), ${\widetilde}{\mathfrak{n}}$ is isomorphic to the nilradical of a Borel subalgebra of the simple Lie algebra with the root 