and ${\mathbb{C}}^{2n}$), and by $x_i$ the coordinate of a vector $x$ corresponding to $e_i$.

The set of all diagonal matrices from ${\mathfrak{g}}$ is a Cartan subalgebra of ${\mathfrak{g}}$; we denote it by ${\mathfrak{h}}$. Let $\Phi$ be the root system of ${\mathfrak{g}}$ with respect to ${\mathfrak{h}}$. Note that $\Phi$ is of type $A_{n-1}$ (respectively, $D_n$, $B_n$ and $C_n$) for ${\mathfrak{sl}}_n({\mathbb{C}})$ (respectively, for ${\mathfrak{so}}_{2n}({\mathbb{C}})$, ${\mathfrak{so}}_{2n+1}({\mathbb{C}})$ and ${\mathfrak{sp}}_{2n}({\mathbb{C}})$). The set of all upper-triangular matrices from ${\mathfrak{g}}$ is a Borel subalgebra of ${\mathfrak{g}}$ containing ${\mathfrak{h}}$; we denote it by ${\mathfrak{b}}$. Let $\Phi^+$ be the set of positive roots with respect to ${\mathfrak{b}}$. As usual, we identify $\Phi^+$ with the following subset of ${\mathbb{R}}^n$: $$\predisplaypenalty=0
\begin{split}
A_{n-1}^+&=\{{\epsilon}_i-{\epsilon}_j,~1\leq i<j\leq n\},\\
B_n^+&=\{{\epsilon}_i-{\epsilon}_j,~1\leq i<j\leq n\}\cup\{{\epsilon}_i+{\epsilon}_j,~1\leq i<j\leq n\}\cup\{{\epsilon}_i,~1\leq i\leq n\},\\
C_n^+&=\{{\epsilon}_i-{\epsilon}_j,~1\leq i<j\leq n\}\cup\{{\epsilon}_i+{\epsilon}_j,~1\leq i<j\leq n\}\cup\{2{\epsilon}_i,~1\leq i\leq n\},\\
D_n^+&=\{{\epsilon}_i-{\epsilon}_j,~1\leq i<j\leq n\}\cup\{{\epsilon}_i+{\epsilon}_j,~1\leq i<j\leq n\}.\\
\end{split}$$ Here $\{{\epsilon}_i\}_{i=1}^n$ is the standard basis of ${\mathbb{R}}^n$.

Denote by ${\mathfrak{n}}$ the algebra of all strictly upper-triangular matrices from ${\mathfrak{g}}$. Then ${\mathfrak{n}}$ has a basis consisting of root vectors $e_{\alpha}$, $\alpha\in\Phi^+$, where $$\predisplaypenalty=0
\begin{split}
e_{{\epsilon}_i}&=\sqrt{2}(e_{0,i}-e_{-i,0}),~e_{2{\epsilon}_i}=e_{i,-i},\\
e_{{\epsilon}_i-{\epsilon}_j}&=\begin{cases}
e_{i,j}&\text{for }A_{n-1},\\
e_{i,j}-e_{-j,-i}&\text{for }B_n,~C_n\text{ and }D_n,
\end{cases}\\
e_{{\epsilon}_i+{\epsilon}_j}&=\begin{cases}
e_{i,-j}-e_{j,-i}&\text{for }B_n\text{ and }D_n,\\
e_{i,-j}+e_{j,-i}&\text{for }C_n,
\end{cases}
\end{split}$$ and $e_{i,j}$ are the usual elementary matrices. For ${\mathfrak{so}}_{2n}({\mathbb{C}})$ (respectively, for ${\mathfrak{so}}_{2n+1}({\mathbb{C}})$ and ${\mathfrak{sp}}_{2n}({\mathbb{C}})$) 