now ready to formulate and prove the main technical proposition needed for the proof of Theorem \[theo:ideal\_fd\_Bn\_Dn\].

**Proposition ..** [*Let, as\[prop:J\_c\_primitive\_Bn\_Dn\] above, $J_c$ be the ideal of $U({\mathfrak{n}})$ generated by ${P}_{\beta}-c_{\beta}$, $\beta\in{\mathcal{B}}$, with $c_{\beta}\neq0$ for $\beta\in{\mathcal{B}}\setminus\Delta$. Then $J_c$ is primitive.*]{}

<span style="font-variant:small-caps;">Proof</span>.

Put $c_1=c_{\beta_1}$. Since $r$ is surjective, $r(J_c)$ is an ideal of $U({\widetilde}{\mathfrak{n}}_e)\otimes{\mathcal{A}}_s$ generated by $r({P}_{\beta})-c_{\beta}$, $\beta\in{\mathcal{B}}\setminus\{\beta_1\}$, or, equivalently, by $e\otimes1-c_{\beta_1}^{-1}c_{\beta_2}$ and by $r({P}_{\beta})-c_{\beta}$, $\beta\in{\widetilde}{\mathcal{B}}={\mathcal{B}}\setminus\{\beta_1,~\beta_2\}$. Note that ${\widetilde}{\mathcal{B}}$ is the Kostant cascade of ${\widetilde}\Phi$ (under the natural identification of ${\widetilde}\Phi$ with the root subsystem $\pm\{\alpha\in\Phi^+\mid{\mathrm{row}}(\alpha)>2\}$ of $\Phi$). Denote by ${\widetilde}{P}_{\beta}$, $\beta\in{\widetilde}\Phi$, the set of canonical generators of $Z({\widetilde}{\mathfrak{n}})$. Our first goal is to check that, up to nonzero scalar, $r({P}_{\beta})$ coincides with ${\widetilde}{P}_{\beta}\otimes1$ for all $\beta\in{\widetilde}{\mathcal{B}}$.

To check this fact, we will use (\[formula:weights\_of\_xi\_i\]). Pick a root $\beta\in{\widetilde}{\mathcal{B}}$. Since $r$ is surjective, $r({P}_{\beta})$ is a central element of $U({\widetilde}{\mathfrak{n}}_e)\otimes{\mathcal{A}}_s$. The center of this algebra has the form $Z({\widetilde}{\mathfrak{n}}_e)\otimes{\mathbb{C}}$, so in fact $r({P}_{\beta})\in Z({\widetilde}{\mathfrak{n}}_e)\otimes{\mathbb{C}}$. The Kostant’s description of the center of the enveloping algebra is valid for the nilradical of a Borel subalgebra of any semisimple Lie algebra with the Kostant cascade being the union of the Kostant cascades of the simple summands [@Kostant1], [@Kostant2]. Note that ${\widetilde}{\mathfrak{n}}_e$ is isomorphic 