that the restriction of the symmetrization map $$\sigma\colon S({\mathfrak{n}})\to U({\mathfrak{n}}),~x^k\mapsto x^k,~x\in{\mathfrak{n}},~k\in{\mathbb{Z}}_{\geq0},$$ to the algebra $S({\mathfrak{n}})^N$ of $N$-invariants is an algebra isomorphism between $S({\mathfrak{n}})^N$ and $Z({\mathfrak{n}})$.

Denote by ${\mathcal{B}}$ the following subset of $\Phi^+$: $${\mathcal{B}}=\begin{cases}\bigcup\nolimits_{1\leq i\leq[n/2]}\{{\epsilon}_i-{\epsilon}_{n-i+1}\}&\text{for }A_{n-1},\\
\bigcup\nolimits_{1\leq i\leq n/2}\{{\epsilon}_{2i-1}-{\epsilon}_{2i},~{\epsilon}_{2i-1}+{\epsilon}_{2i}\}&\text{for }B_n,~n\text{ even},\\
\bigcup\nolimits_{1\leq i\leq[n/2]}\{{\epsilon}_{2i-1}-{\epsilon}_{2i},~{\epsilon}_{2i-1}+{\epsilon}_{2i}\}\cup\{{\epsilon}_n\}&\text{for }B_n,~n\text{ odd},\\
\bigcup\nolimits_{1\leq i\leq n}\{2{\epsilon}_i\}&\text{for }C_n,\\
\bigcup\nolimits_{1\leq i\leq[n/2]}\{{\epsilon}_{2i-1}-{\epsilon}_{2i},~{\epsilon}_{2i-1}+{\epsilon}_{2i}\}&\text{for }D_n.\\
\end{cases}$$ Note that ${\mathcal{B}}$ is a maximal strongly orthogonal subset of $\Phi^+$, i.e., ${\mathcal{B}}$ is maximal with the property that if $\alpha,\beta\in{\mathcal{B}}$ then neither $\alpha-\beta$ nor $\alpha+\beta$ belongs to $\Phi^+$. We call ${\mathcal{B}}$ the *Kostant cascade* of orthogonal roots in $\Phi^+$.

We next present a canonical set of generators of $Z({\mathfrak{n}})$ (or, equivalently, of $S({\mathfrak{n}})^N$), whose description goes back to J. Dixmier, A. Joseph and B. Kostant [@Dixmier1], [@Joseph1], [@Kostant1], [@Kostant2]. We can consider ${\mathbb{Z}}\Phi$, the ${\mathbb{Z}}$-linear span of $\Phi$, as a subgroup of the group ${\mathfrak{X}}$ of rational multiplicative characters of $H$ by putting $\pm{\epsilon}_i(h)=h_{i,i}^{\pm1}$, where $h_{i,i}$ is the $i$-th diagonal element of a matrix $h\in H$. Recall that a vector $\lambda\in{\mathbb{R}}^n$ is called a *weight* of $H$ if $c(\alpha,\lambda)=2(\alpha,\lambda)/(\alpha,\alpha)$ is an integer for any $\alpha\in\Phi^+$, where $(\cdot,\cdot)$ is the standard inner product on ${\mathbb{R}}^n$. A weight $\lambda$ is called *dominant* if $c(\alpha,\lambda)\geq0$ for all $\alpha\in\Phi^+$. An 