$$\xi_{{\epsilon}_i-{\epsilon}_{n-i+1}}=\begin{vmatrix}\label{formula:Delta_i_A_n}
e_{1,n-i+1}&\ldots&e_{1,n-1}&e_{1,n}\\
e_{2,n-i+1}&\ldots&e_{2,n-1}&e_{2,n}\\
\vdots&\iddots&\vdots&\vdots\\
e_{i,n-i+1}&\ldots&e_{i,n-1}&e_{i,n}\\
\end{vmatrix}=\begin{vmatrix}
e_{{\epsilon}_1-{\epsilon}_{n-i+1}}&\ldots&e_{{\epsilon}_1-{\epsilon}_{n-1}}&e_{{\epsilon}_1-{\epsilon}_n}\\
e_{{\epsilon}_2-{\epsilon}_{n-i+1}}&\ldots&e_{{\epsilon}_2-{\epsilon}_{n-1}}&e_{{\epsilon}_2-{\epsilon}_n}\\
\vdots&\iddots&\vdots&\vdots\\
e_{{\epsilon}_i-{\epsilon}_{n-i+1}}&\ldots&e_{{\epsilon}_i-{\epsilon}_{n-1}}&e_{{\epsilon}_i-{\epsilon}_n}\\
\end{vmatrix}.$$

ii\) $\Phi=C_n$. Here, for $1\leq i\leq m=n$, $$\xi_{2{\epsilon}_i}=\begin{vmatrix}
e_{{\epsilon}_1+{\epsilon}_i}&\ldots&e_{{\epsilon}_1+{\epsilon}_3}&e_{{\epsilon}_1+{\epsilon}_2}&2e_{2{\epsilon}_1}\\
e_{{\epsilon}_2+{\epsilon}_i}&\ldots&e_{{\epsilon}_2+{\epsilon}_3}&2e_{2{\epsilon}_2}&e_{{\epsilon}_1+{\epsilon}_2}\\
e_{{\epsilon}_3+{\epsilon}_i}&\ldots&2e_{2{\epsilon}_3}&e_{{\epsilon}_2+{\epsilon}_3}&e_{{\epsilon}_1+{\epsilon}_3}\\
\vdots&\iddots&\vdots&\vdots&\vdots\\
2e_{2{\epsilon}_i}&\ldots&e_{{\epsilon}_3+{\epsilon}_i}&e_{{\epsilon}_2+{\epsilon}_i}&e_{{\epsilon}_1+{\epsilon}_i}\\
\end{vmatrix}.\label{formula:Delta_i_C_n}$$

iii\) $\Phi=D_n$. If $i$ is odd, then $$\xi_{{\epsilon}_i+{\epsilon}_{i+1}}^2=\pm\begin{vmatrix}
e_{{\epsilon}_1+{\epsilon}_{i+1}}&\ldots&e_{{\epsilon}_1+{\epsilon}_3}&e_{{\epsilon}_1+{\epsilon}_2}&0\\
e_{{\epsilon}_2+{\epsilon}_{i+1}}&\ldots&e_{{\epsilon}_2+{\epsilon}_3}&0&-e_{{\epsilon}_1+{\epsilon}_2}\\
e_{{\epsilon}_3+{\epsilon}_{i+1}}&\ldots&0&-e_{{\epsilon}_2+{\epsilon}_3}&-e_{{\epsilon}_1+{\epsilon}_3}\\
\vdots&\iddots&\vdots&\vdots&\vdots\\
0&\ldots&-e_{{\epsilon}_3+{\epsilon}_{i+1}}&-e_{{\epsilon}_2+{\epsilon}_{i+1}}&-e_{{\epsilon}_1+{\epsilon}_{i+1}}\\
\end{vmatrix}.\label{formula:Delta_i_D_n_pf}\postdisplaypenalty=10000$$ (After suitable reordering of indices, the matrix in the right-hand side becomes skew-symmetric, so $\xi_{{\epsilon}_i+{\epsilon}_{i+1}}$ is nothing but its Pfaffian.) Our normalization is such that the term $e_{{\epsilon}_1+{\epsilon}_2}e_{{\epsilon}_3+{\epsilon}_4}\ldots e_{{\epsilon}_i+{\epsilon}_{i+1}}$ enters $\xi_{{\epsilon}_i+{\epsilon}_{i+1}}$ with coefficient 1.

Next, assume that $i\leq n-1$ is even. Let ${\mathcal{U}}^i$ be the $(i\times i)$-matrix with entries from $S({\mathfrak{n}})$ defined by $({\mathcal{U}}^i)_{a,b}=-({\mathcal{U}}^i)_{i-b+1,i-a+1}=e_{{\epsilon}_a+{\epsilon}_{i-b+1}}$ for $a<i-b+1$, $({\mathcal{U}}^i)_{a,i-a+1}=0$, and let ${\mathcal{U}}_s^i$ be the matrix obtained from ${\mathcal{U}}^i$ by deleting the $(i-s+1)$-th row and column. Then we can set $$\begin{split}
a_s&=\sum\limits_{j=s+1}^ne_{{\epsilon}_s-{\epsilon}_j}\label{formula:Delta_i_D_n_net}e_{{\epsilon}_s+{\epsilon}_j},\\
\xi_{{\epsilon}_{i-1}-{\epsilon}_i}&=\sum\limits_{s=1}^ia_s\det{\mathcal{U}}_s^i.\\
\end{split}$$ Finally, assume $m=n$ is even. In this case $\xi_{{\epsilon}_{n-1}-{\epsilon}_n}$ can be defined by $$\xi_{{\epsilon}_{n-1}-{\epsilon}_n}^2=\pm\begin{vmatrix}
e_{{\epsilon}_1-{\epsilon}_n}&e_{{\epsilon}_1+{\epsilon}_{n-1}}&\ldots&e_{{\epsilon}_1+{\epsilon}_2}&0\\
e_{{\epsilon}_2-{\epsilon}_n}&e_{{\epsilon}_2+{\epsilon}_{n-1}}&\ldots&0&-e_{{\epsilon}_1+{\epsilon}_2}\\
\vdots&\vdots&\iddots&\vdots&\vdots\\
e_{{\epsilon}_{n-1}-{\epsilon}_n}&0&\ldots&-e_{{\epsilon}_2+{\epsilon}_{n-1}}&-e_{{\epsilon}_1+{\epsilon}_{n-1}}\\
0&-e_{{\epsilon}_{n-1}-{\epsilon}_n}&\ldots&-e_{{\epsilon}_2-{\epsilon}_n}&-e_{{\epsilon}_1-{\epsilon}_n}\\
\end{vmatrix}$$ (our normalization is such that the term $e_{{\epsilon}_{n-1}-{\epsilon}_n}\xi_{{\epsilon}_{n-3}+{\epsilon}_{n-2}}$ enters $\xi_{{\epsilon}_{n-1}+{\epsilon}_n}$ with coefficient 1).

iv\) $\Phi=B_n$. If $i\leq n-1$ is odd, then $\xi_{{\epsilon}_i+{\epsilon}_{i+1}}$ can be defined via formula (\[formula:Delta\_i\_D\_n\_pf\]). Next, for even $i$, we can define $\xi_{{\epsilon}_i-{\epsilon}_{i+1}}$ via formula (\[formula:Delta\_i\_D\_n\_net\]) with $$b_s=\sum\nolimits_{j=s+1}^ne_{{\epsilon}_s-{\epsilon}_j}e_{{\epsilon}_s+{\epsilon}_j}+e_{{\epsilon}_s}^2/4$$ instead of $a_s$. Finally, assume $n=m$ is odd. Then $\xi_{{\epsilon}_n}$ can be defined by $$\xi_{{\epsilon}_n}^2=\pm\begin{vmatrix}
e_{{\epsilon}_1}&e_{{\epsilon}_1+{\epsilon}_n}&\ldots&e_{{\epsilon}_1+{\epsilon}_2}&0\\
e_{{\epsilon}_2}&e_{{\epsilon}_2+{\epsilon}_n}&\ldots&0&-e_{{\epsilon}_1+{\epsilon}_2}\\
\vdots&\vdots&\iddots&\vdots&\vdots\\
e_{{\epsilon}_n}&0&\ldots&-e_{{\epsilon}_2+{\epsilon}_n}&-e_{{\epsilon}_1+{\epsilon}_n}\\
