the proof of [@IgnatyevPenkov1 Theorem 3.1], define inductively two (finite) sequences $\{i_j\}$ and $\{k_j\}$ of positive integers by the following rule. If $i_j$ is already defined and there exists $k>i_j$ such that $c_{\beta}\neq0$ for $\beta\in{\mathcal{B}}\cap{\mathcal{R}}_k$ with ${\mathrm{col}}(\beta)\leq0$, then set $k_j$ to be the minimal among all such $k$. Similarly, if $k_j$ is already defined and there exists $i>k_j$ such that $c_{\beta}=0$ for $\beta\in({\mathcal{B}}\setminus\Delta)\cap{\mathcal{R}}_i$, then set $i_{j+1}$ to be the minimal among all such $i$.

To each $j$ for which both $i_j$ and $k_j$ exist we assign the set of roots $$\begin{split}
\Gamma_j&=\{{\epsilon}_{i_j}+{\epsilon}_{i_j+2},~{\epsilon}_{i_j}-{\epsilon}_{i_j+2}\}\cup\bigcup\limits_{i_j<s<k_j-1,~s\text{ even}}\{{\epsilon}_s+{\epsilon}_{s+3},~{\epsilon}_s-{\epsilon}_{s+3}\}\cup\Gamma_j',\text{ where}\\
\Gamma_j'&=\begin{cases}
\{{\epsilon}_{n-1}\},&\text{if $\Phi=B_n$ and $k_j=n$},\\
\{{\epsilon}_{k_j-1}+{\epsilon}_{k_j+1},~{\epsilon}_{k_j-1}-{\epsilon}_{k_j+1}\}&\text{otherwise}.
\end{cases}
\end{split}$$ Denote the length of the sequence $\{i_j\}$ by $l_I$, then the length $l_K$ of the sequence $\{k_J\}$ is either $l_I$ or $l_I-1$. If $i_j$ exists and $k_j$ does not exist (i.e., if $l_K=l_I-1$ and $j=l_I$), then put $$\begin{split}
\Gamma_j&=\{{\epsilon}_{i_j}+{\epsilon}_{i_j+2},~{\epsilon}_{i_j}-{\epsilon}_{i_j+2}\}\cup\bigcup\limits_{i_j<s<n-2,~s\text{ even}}\{{\epsilon}_s+{\epsilon}_{s+3},~{\epsilon}_s-{\epsilon}_{s+3}\}\cup\Gamma_j',\text{ where}\\
\Gamma_j'&=\begin{cases}
\{{\epsilon}_{n-2}\},&\text{if $\Phi=B_n$, $n$ even},\\
\{{\epsilon}_{n-2}-{\epsilon}_n,~{\epsilon}_{k_j-1}-{\epsilon}_{k_j+1}\},&\text{if $\Phi=D_n$, $n$ even},\\
\varnothing&\text{otherwise}.
\end{cases}
\end{split}$$ Finally, put $$X=\left({\mathcal{B}}\cup\bigcup_{j=1}^{l_I}\Gamma_j\right)\setminus\left(\bigcup\{{\mathcal{R}}_i\mid c_{\beta}=0\text{ for }\beta\in({\mathcal{B}}\setminus\Delta)\cap{\mathcal{R}}_i\}\cup\bigcup_{j=1}^{l_K}{\mathcal{R}}_{k_j}\right).$$

Given a map ${\varphi}\colon X\to{\mathbb{C}}$, we denote $$\mu_{{\varphi}}=\sum_{\alpha\in X}{\varphi}(\alpha)e_{\alpha}^*.$$ By [@Dixmier3 6.6.9 (c)], ${P}_{\beta}-\xi_{\beta}(\mu_{{\varphi}})\in J(\mu_{{\varphi}})$, $\beta\in{\mathcal{B}}$. Using explicit formulas for ${P}_{\beta}$’s from Subsection \[sst:center\_fd\], one can easily construct a map ${\varphi}_1\colon X\to{\mathbb{C}}$ such that ${\varphi}_1(\beta)\neq0$ if ${\mathrm{col}}(\beta)\leq0$, and $\xi_{\beta}(\mu_{{\varphi}_1})=c_{\beta}$ 