$k\geq 1$. Here $\beta_k={\epsilon}_k-{\epsilon}_{n-k+1}$ for $\Phi=A_{n-1}$ and $\beta_k=2{\epsilon}_k$ for $\Phi=C_n$.*]{}

One of the key ingredients in the proof of this result is to check that if condition (ii) is satisfied then $J_c$ is primitive. To do this, in [@IgnatyevPenkov1] an explicit set of generators of the quotient algebra $U({\mathfrak{n}})/J_c$ was constructed. It turns out that these generators satisfy (up to scalars) the defining relations of the Weyl algebra ${\mathcal{A}}_s$ for $s=|\Phi^+\setminus{\mathcal{B}}|/2$. Since ${\mathcal{A}}_s$ is simple and, as one can easily check, $J_c\neq U({\mathfrak{n}})$, we conclude that $U({\mathfrak{n}})/J_c\cong{\mathcal{A}}_s$, and, consequently, $J_c$ is primitive. Below we briefly recall the explicit formulas for these generators from [@IgnatyevPenkov1 Subsection 3.1].

We define the maps ${\mathrm{row}}\colon\Phi^+\to{\mathbb{Z}}$ and ${\mathrm{col}}\colon\Phi^+\to{\mathbb{Z}}$ by putting ${\mathrm{row}}({\epsilon}_i-{\epsilon}_j)={\mathrm{row}}({\epsilon}_i+{\epsilon}_j)={\mathrm{row}}(2{\epsilon}_i)={\mathrm{row}}({\epsilon}_i)=i$, ${\mathrm{col}}({\epsilon}_i+{\epsilon}_j)={\mathrm{col}}(2{\epsilon}_j)=-j$. Let ${\mathcal{R}}_i=\{\alpha\in\Phi^+\mid{\mathrm{row}}(\alpha)=i\}$. For $\alpha\in\Phi^+$, where $\Phi=A_{n-1}$ or $C_n$, set $$\begin{split}
A(\alpha)&=\begin{cases}\bigcup\nolimits_{j+1\leq k\leq n-i+1}\{{\epsilon}_j-{\epsilon}_k\},&\text{if $\Phi=A_{n-1}$, $\alpha={\epsilon}_i-{\epsilon}_j$, $j<n-i+1$},\\
\bigcup\nolimits_{n-j+1\leq k\leq i-1}\{{\epsilon}_k-{\epsilon}_i\},&\text{if $\Phi=A_{n-1}$, $\alpha={\epsilon}_i-{\epsilon}_j$, $j>n-i+1$},\\
\bigcup\nolimits_{i\leq k\leq j-1}\{{\epsilon}_k+{\epsilon}_j\}\cup{\mathcal{R}}_j,&\text{if $\Phi=C_n$, $\alpha={\epsilon}_i-{\epsilon}_j$},\\
\bigcup\nolimits_{i\leq k\leq j-1}\{{\epsilon}_k-{\epsilon}_j\},&\text{if $\Phi=C_n$, $\alpha={\epsilon}_i+{\epsilon}_j$},\\
\end{cases}\\
{\mathcal{B}}(\alpha)&=\{\alpha\}\cup\{\beta\in{\mathcal{B}}\mid{\mathrm{row}}(\beta)<{\mathrm{row}}(\alpha)\},\\
R(\alpha)&=\{{\mathrm{row}}(\gamma),~\gamma\in{\mathcal{B}}(\alpha)\},~C(\alpha)=\{{\mathrm{col}}(\gamma),~\gamma\in{\mathcal{B}}(\alpha)\}.
\end{split}$$ Define a matrix ${\mathcal{U}}$ with entries from $U({\mathfrak{n}})$ by the following rule.

  $\Phi$      Size of ${\mathcal{U}}$   ${\mathcal{U}}$
  ----------- ------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  $A_{n-1}$   $n\times n$               ${\mathcal{U}}_{i,j}=e_{{\epsilon}_i-{\epsilon}_j}$ for $1\leq i<j\leq n$,
                                        ${\mathcal{U}}_{i,j}=0$ otherwise
  $C_n$       $2n\times2n$              ${\mathcal{U}}_{i,j}=-{\mathcal{U}}_{-j,-i}=e_{{\epsilon}_i-{\epsilon}_j}$, ${\mathcal{U}}_{i,-j}={\mathcal{U}}_{j, -i}=e_{{\epsilon}_i+{\epsilon}_j}$ for $1\leq i<j\leq n$,
                                        ${\mathcal{U}}_{i,-i}=2e_{{\epsilon}_{2i}}$, $1\leq i\leq n$, ${\mathcal{U}}_{i,j}=0$ otherwise

As before, 