n$,
                                     ${\mathcal{U}}_{i,j}=0$ otherwise

Denote by $\Delta_{\alpha}$ the element of $U({\mathfrak{n}})$, which equals the minor of the matrix ${\mathcal{U}}$ with the set of rows $R(\alpha)$ and the set of columns $C(\alpha)$. Note that variables in each summand of each $\Delta_{\alpha}$ commute. For instance, if $\Phi=D_n$, $n\geq6$, and $\alpha={\epsilon}_4-{\epsilon}_6$, then $$\Delta_{\alpha}=\begin{vmatrix}
e_{{\epsilon}_1-{\epsilon}_6}&e_{{\epsilon}_1+{\epsilon}_4}&e_{{\epsilon}_1+{\epsilon}_2}&0\\
e_{{\epsilon}_2-{\epsilon}_6}&e_{{\epsilon}_2+{\epsilon}_4}&0&-e_{{\epsilon}_1+{\epsilon}_2}\\
e_{{\epsilon}_3-{\epsilon}_6}&e_{{\epsilon}_3+{\epsilon}_4}&-e_{{\epsilon}_2+{\epsilon}_3}&-e_{{\epsilon}_1+{\epsilon}_3}\\
e_{{\epsilon}_4-{\epsilon}_6}&0&-e_{{\epsilon}_2+{\epsilon}_4}&-e_{{\epsilon}_1+{\epsilon}_4}\\
\end{vmatrix}.$$

**Lemma ..** [*Let $\alpha\in\Phi^+\setminus{\mathcal{B}}$ and $\gamma\in\Phi^+$. If $\gamma\notin A(\alpha)$ then $[\Delta_{\alpha},e_{\gamma}]=0$\[lemm:comm1\_B\_n\_D\_n\]. If $\gamma\in A(\alpha)\setminus\Delta$ then $\alpha+\gamma\in\Phi^+$ and $[\Delta_{\alpha},e_{\gamma}]=\pm\Delta_{\alpha+\gamma}$.*]{}

<span style="font-variant:small-caps;">Proof</span>. Let $\Phi=D_n$, $\alpha={\epsilon}_i-{\epsilon}_j$, $i$ is even (all other cases can be considered similarly). One has $$\Delta_{\alpha}=\begin{vmatrix}
e_{{\epsilon}_1-{\epsilon}_j}&e_{{\epsilon}_1+{\epsilon}_i}&e_{{\epsilon}_1+{\epsilon}_{i-2}}&e_{{\epsilon}_1+{\epsilon}_{i-3}}&\ldots&
e_{{\epsilon}_1+{\epsilon}_3}&e_{{\epsilon}_1+{\epsilon}_2}&0\\
e_{{\epsilon}_2-{\epsilon}_j}&e_{{\epsilon}_2+{\epsilon}_i}&e_{{\epsilon}_2+{\epsilon}_{i-2}}&e_{{\epsilon}_2+{\epsilon}_{i-3}}&\ldots&
e_{{\epsilon}_2+{\epsilon}_3}&0&-e_{{\epsilon}_1+{\epsilon}_2}\\
e_{{\epsilon}_3-{\epsilon}_j}&e_{{\epsilon}_3+{\epsilon}_i}&e_{{\epsilon}_3+{\epsilon}_{i-2}}&e_{{\epsilon}_3+{\epsilon}_{i-3}}&\ldots&
0&-e_{{\epsilon}_2+{\epsilon}_3}&-e_{{\epsilon}_1+{\epsilon}_3}\\
e_{{\epsilon}_4-{\epsilon}_j}&e_{{\epsilon}_4+{\epsilon}_i}&e_{{\epsilon}_4+{\epsilon}_{i-2}}&e_{{\epsilon}_4+{\epsilon}_{i-3}}&\ldots&
-e_{{\epsilon}_3+{\epsilon}_4}&-e_{{\epsilon}_2+{\epsilon}_4}&-e_{{\epsilon}_1+{\epsilon}_4}\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\
e_{{\epsilon}_{i-2}-{\epsilon}_j}&e_{{\epsilon}_{i-2}+{\epsilon}_i}&0&-e_{{\epsilon}_{i-3}+{\epsilon}_{i-2}}&\ldots&
-e_{{\epsilon}_3+{\epsilon}_{i-2}}&-e_{{\epsilon}_2+{\epsilon}_{i-2}}&-e_{{\epsilon}_1+{\epsilon}_{i-2}}\\
e_{{\epsilon}_{i-1}-{\epsilon}_j}&e_{{\epsilon}_{i-1}+{\epsilon}_i}&-e_{{\epsilon}_{i-2}+{\epsilon}_{i-1}}&-e_{{\epsilon}_{i-3}+{\epsilon}_{i-1}}&\ldots&
-e_{{\epsilon}_3+{\epsilon}_{i-1}}&-e_{{\epsilon}_2+{\epsilon}_{i-1}}&-e_{{\epsilon}_1+{\epsilon}_{i-1}}\\
e_{{\epsilon}_i-{\epsilon}_j}&0&-e_{{\epsilon}_{i-2}+{\epsilon}_i}&-e_{{\epsilon}_{i-3}+{\epsilon}_i}&\ldots&
-e_{{\epsilon}_3+{\epsilon}_3}&-e_{{\epsilon}_2+{\epsilon}_i}&-e_{{\epsilon}_1+{\epsilon}_i}\\
\end{vmatrix}.$$

Suppose that $e_{\gamma}$, $\gamma\in\Phi^+$, does not commute with one of $e_{\delta}$’s involved in $\Delta_{\alpha}$. If $\gamma={\epsilon}_j+{\epsilon}_k$ for some $k>j$ then, clearly, $[\Delta_{\alpha},e_{\gamma}]$ equals the minor obtained from $\Delta_{\alpha}$ by replacing all $e_{{\epsilon}_s-{\epsilon}_j}$ from the first column by ${\epsilon}_s+{\epsilon}_k$, but it is exactly $\Delta_{{\epsilon}_i+{\epsilon}_k}$. If $\gamma={\epsilon}_k+{\epsilon}_j$ for some $i<k<j$ then $[\Delta_{\alpha},e_{\gamma}]$ equals the minor obtained from $\Delta_{\alpha}$ by replacing all $e_{{\epsilon}_s-{\epsilon}_j}$ from the first column by $-{\epsilon}_s+{\epsilon}_k$, but this minor is nothing but $-\Delta_{{\epsilon}_i+{\epsilon}_k}$. One can argue similarly for $\gamma={\epsilon}_{i-1}+{\epsilon}_j$. In fact, we considered all possibilities for $\gamma\in A(\alpha)\setminus\Delta$.

Next, if $\gamma={\epsilon}_k+{\epsilon}_j$ for some $k<i-1$, 