tcb@breakable

Superintegrable systems associated to commutants of Cartan subalgebras in
enveloping algebras

Rutwig Campoamor-Stursberg1,⋆, Danilo Latini2,∗, Ian Marquette3,∙, Junze
Zhang4,† and Yao-Zhong Zhang4,‡

1 Instituto de Matemática Interdisciplinar and Dpto. Geometría y Topología,
UCM, E-28040 Madrid, Spain

2 Via Appia Antica 2, 00045 Genzano di Roma RM, Italy

${}^{1}3$ Department of Mathematical and Physical Sciences, La Trobe
University, Bendigo, VIC 3552, Australia

4 School of Mathematics and Physics, The University of Queensland, Brisbane,
QLD 4072, Australia

<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract.

We provide a classification and explicit formulas for the elements that span
the centralizer of Cartan subalgebras of complex semisimple Lie algebras of
non-exceptional type in their universal enveloping algebra, and show that
these generate polynomial (Poisson) algebras. The precise structure for the
case of rank three simple Lie algebras is provided, and the inclusion
relations between the corresponding polynomial algebras illustrated. We
develop the ideas of constructing algebraic superintegrable systems and their
integrals from the generators of the centralizer subalgebras and their
symmetry algebra. In particular, we present explicitly one example the
superintegrable system corresponding to the Cartan centralizer of
$\mathfrak{so}(6,\mathbb{C})$.

##  1\. Introduction

Superintegrable and integrable systems occupy a pivotal position within
mathematical physics and symplectic geometry, due to their rich algebraic and
geometric structure, as well as their wide applications [19, 27]. They further
provide a natural connection with the theory of special functions and
orthogonal polynomials, in particular, the Askey-Wilson scheme and the
Painlevé transcendent. It is well known that superintegrable systems are
intimately related to finitely generated Poisson algebras [13, 14, 15, 32,
33], a structure that has been observed and exploited in many problems of
mathematical physics [3, 31]. The study of classical and quantum
superintegrable systems, from a mathematical perspective, can be traced back
to the 1960s [19], while the description of classical superintegrable systems
in its modern form was first introduced in [28]. These systems generalize
Liouville-integrable Hamiltonian systems on a $2n$-dimensional symplectic
manifold in the case where isotropic (invariant) submanifolds have dimension
less than $n$. In the classical case, a superintegrable system is generally
considered as a deformation of the Poisson centers of a symplectic manifold
[33]. That is, the disjoint union of symplectic leaves in the phase space
forms a non-commutative algebra via Poisson projections. By making the phase
space explicit, it can also be constructed through the embedding chain of some
associative algebras [34]. The problem of classifying $n$-dimensional
quadratically (i.e. only second order integrals) superintegrable systems was
investigated very recently using new geometrical approaches in [24]. The paper
here, will not restrict to quadratic integrals, and will develop algebraic
approaches based on commutants and classical Lie algebras. The construction of
superintegrable systems based on Lie algebras is known to be deeply related to
their invariants (generalized Casimir invariants), as the co-adjoint orbits of
dual Lie algebras can be naturally viewed as a symplectic leaf [18, 26]. Such
systems are important for understanding the structural properties of Calogero-
Moser and Ruijsenaars systems, among others. On the other hand, by lifting a
Lie algebra to its universal enveloping algebra, many studies reveal that non-
commutative polynomial (Poisson) structures are hidden in the centralizer of
certain subalgebras of enveloping algebras [6, 7, 8, 9, 10, 20]. Concerning
explicit models, purely algebraic constructions of superintegrable systems
over enveloping algebras of finite-dimensional Lie algebras have been studied
for many different hierarchies of systems. For instance, this approach was
successfully implemented for the Lie algebra $\mathfrak{su}(3)$ [11] and
$\mathfrak{gl}(3)$ [8], by applying an algorithm on centralizers of
subalgebras. More recent examples, which connect with different context of
mathematical physics can be found in [7, 9, 12, 21]. In particular, in a
recent paper [6], a similar algorithm was also applied to the Cartan
centralizer of $A_{n}$ and certain non-semisimple algebras [9]. These
polynomial Poisson algebras associated with the centralizer of Cartan
subalgebras can all be realized in terms of high-rank Racah-type algebras. In
this paper, we extend the results of [6] to the remaining classical semisimple
Lie algebras, and we provide the classification of generators for the Cartan
centralizers via their respective root system. The purpose is to characterize
the related polynomials completely and explicitly.

The structure of this paper is as follows. In Section 2, we review the formal
definition of commutants, i.e., the centralizer of a subalgebra in the
enveloping algebra of a finite dimensional Lie algebra. In order to find
explicit formulas of invariant homogeneous polynomials, we provide an
algorithm and define polynomial (Poisson) algebras in commutants (or
centralizers). This method, as mentioned, has been found to be of interest in
the construction of superintegrable systems in enveloping algebras (see, e.g.
[6] and references therein). In Section $\ref{3},$ we construct a linear basis
for a vector subspace in Cartan commutants, and define a proper Poisson type
bracket on that vector space such that it generates a polynomial algebra. We
call such algebras the polynomial algebras generated from Cartan centralizers.
In Section 4, using an ordered basis that satisfies the Serre-Chevalley
relations, we provide explicit formulas of generators in the polynomial
algebras $\mathcal{Q}_{B_{n}}(2n-1)$, $\mathcal{Q}_{C_{n}}(2n-1)$ and
$\mathcal{Q}_{D_{n}}(2n-2),$ where $B_{n},C_{n}$ and $D_{n}$ are classical
semisimple Lie algebras for $n\geq 1.$. The inclusion relations and degrees of
these polynomial algebras are also given. The formal proof of the
classification is rather nontrivial, and details concerning it will be given
in a forthcoming paper [5]. As an illustration of the classification, we
compute the indecomposable homogeneous generators in the Cartan centralizers
for rank three semisimple Lie algebras of non-exceptional types. In
particular, generators and algebraic structures of
$\mathcal{Q}_{B_{3}}(5),\mathcal{Q}_{C_{3}}(5)$ and $\mathcal{Q}_{D_{3}}(3)$,
which is itself of interest, albeit $D_{3}$ is isomorphic to $A_{3}$. The
dimension of these polynomial algebras and the inclusion relation will also be
given in Subsection 5.1, 5.3 and 5.2 respectively. In particular, we have also
studied the superintegrable system corresponding to the polynomial algebra
$\mathcal{Q}_{D_{3}}(3).$ Finally, in Section 6, some conclusions and future
perspectives will be drawn.

##  2\. Preliminaries

Let $\mathfrak{g}$ be a $n$-dimensional semisimple or reductive Lie algebra
over a field $\mathbb{F}$ of characteristic zero with commutator
$[\cdot,\cdot]_{o}$ and let $\beta_{\mathfrak{g}}=\\{X_{1},\ldots,X_{n}\\}$ an
ordered basis. Non-trivial brackets satisfy the relations
$[X_{i},X_{j}]_{o}=C_{ij}^{k}X_{k},$ where $\left\\{C_{ij}^{k}\right\\}$
denote the structure constants of $\mathfrak{g}$ over the given basis. Let
$\mathfrak{g}^{*}$ denote the dual space of $\mathfrak{g}$, which admits a
canonical bracket $\\{\cdot,\cdot\\}_{o}$ with coordinates
$\\{x_{1},\ldots,x_{n}\\}$ determined by a dual basis of
$\beta_{\mathfrak{g}}$. Let $U(\mathfrak{g})$ be the universal enveloping
algebra of $\mathfrak{g}$. As is well-known [17], the symmetric algebra
$S(\mathfrak{g})\cong\mathbb{F}[\mathfrak{g}^{*}]$ of $\mathfrak{g}$ is
endowed with the Poisson-Lie bracket
$\\{\cdot,\cdot\\}_{o}:S(\mathfrak{g})\times S(\mathfrak{g})\rightarrow
S(\mathfrak{g})$, defined by

(1)
$\displaystyle\\{p,q\\}_{o}=\sum_{j,k,l=1}^{n}C_{jk}^{l}x_{l}\dfrac{\partial
p}{\partial x_{j}}\dfrac{\partial q}{\partial x_{k}},\quad\forall p,q\in
S(\mathfrak{g}),$

which defines a Poisson algebra
$\left(S(\mathfrak{g}),\\{\cdot,\cdot\\}_{o}\right).$ For more details of the
construction of $\\{\cdot,\cdot\\}_{o}$ on $S(\mathfrak{g})$, see, for
instance, [29, Chapter 6, Section 6.1].

In the following, we are interested in the action of the adjoint and coadjoint
orbits of $\mathfrak{g}$ on the enveloping algebra and symmetric algebra,
respectively. We recall that the adjoint representation
$ad:\mathfrak{g}\rightarrow Aut(U(\mathfrak{g}))$ and the coadjoint
representation $ad^{*}:\mathfrak{g}\rightarrow
Aut\left(S(\mathfrak{g})\right)$ induce the following actions [22]

(2) $\displaystyle\quad P\left(X_{1},\ldots,X_{n}\right)\in
U(\mathfrak{g})\mapsto[X_{j},P]_{o}\in U(\mathfrak{g}),$ (3)
$\displaystyle\quad p(x_{1},\ldots,x_{n})\in
S(\mathfrak{g})\mapsto\\{x_{j},p\\}_{o}=\bar{X}_{j}(p(x_{1},\ldots,x_{n}))=\sum_{l,k}C_{jk}^{l}x_{l}\dfrac{\partial
p}{\partial x_{k}}\in S(\mathfrak{g}),$

where $\bar{X}_{j}=\sum_{l,k}C_{jk}^{l}x_{l}\dfrac{\partial}{\partial x_{k}}$
is the first-order differential operator associated to the generator $X_{j}$
of $\mathfrak{g}.$ Using the Leibniz rule, it can be shown by recursion that
for any $\prod_{i}f_{i},\prod_{j}g_{j}$ with $f_{i},g_{j}\in S(\mathfrak{g})$,
the following identity is satisfied:

(4)
$\displaystyle\left\\{\prod_{i=1}^{m}f_{i},\prod_{j=1}^{n}g_{j}\right\\}_{o}=\sum_{i_{s},j_{r}}\\{f_{i_{s}},g_{j_{r}}\\}_{o}\prod_{i\neq
i_{s}}f_{i}\prod_{j\neq j_{r}}g_{j}.$

This formula, when applied to monomials, provides a systematic procedure to
compute the Poisson bracket of (homogeneous) polynomials in systematic way.

We now turn to the notion of commutant relative to a subalgebra within the
universal enveloping algebra and the symmetric algebra of a finite dimensional
Lie algebra $\mathfrak{g}$, respectively.

######  Definition 2.1.

Let $\mathfrak{a}$ be a subalgebra of $\mathfrak{g}$ with an ordered basis
$\beta_{\mathfrak{a}}=\\{X_{1},\ldots,X_{s}\\}$, and let $\mathfrak{a}^{*}$ be
its dual. The commutant (or centralizer subalgebra)
$U(\mathfrak{g})^{\mathfrak{a}}$ and $S(\mathfrak{g})^{\mathfrak{a}}$ is
respectively defined as the centralizer of $\mathfrak{a}$ and
$\mathfrak{a}^{*}$ in $U(\mathfrak{g})$ and $S(\mathfrak{g})$:

$\displaystyle U(\mathfrak{g})^{\mathfrak{a}}=$ $\displaystyle\left\\{P\in
U(\mathfrak{g}):\text{ }[X,P]_{o}=0\quad\forall X\in\mathfrak{a}\right\\},$
$\displaystyle S(\mathfrak{g})^{\mathfrak{a}}=$ $\displaystyle\left\\{p\in
S(\mathfrak{g}):\text{ }\\{x,p\\}_{o}=0\quad\forall
x\in\mathfrak{a}^{*}\right\\},$

where $P$ and $p$ are polynomials in terms of $X_{j}$ and $x_{j}$ for all
$1\leq j\leq n$.

######  Remark 2.2.

The Poisson centre of $\left(S(\mathfrak{g}),\\{\cdot,\cdot\\}_{o}\right)$ is
the set of all $\mathfrak{g}$-invariant polynomials, i.e.,

$\displaystyle S(\mathfrak{g})^{\mathfrak{g}}=\left\\{p\in
S(\mathfrak{g}):\text{ }\\{p,x\\}_{o}=0\quad\forall
x\in\mathfrak{g}^{*}\right\\}.$

For any $h\in\mathbb{N}$, we define

$\mathcal{U}_{h}(\mathfrak{g})=\\{X_{1}^{i_{1}}\ldots
X_{n}^{i_{n}}:i_{1}+\ldots+i_{n}\leq h\\}$

as the linear subspace of $\mathcal{U}(\mathfrak{g})$ spanned by polynomials
of degree at most $h$ in the (noncommutative) generators of $\mathfrak{g}$.
The degree $\delta$ of an arbitrary element $P\in\mathcal{U}(\mathfrak{g})$ is
defined as $\delta={\rm inf}\\{k:P\in\mathcal{U}_{k}(\mathfrak{g})\\}$.

Furthermore, there is a natural filtration in $U(\mathfrak{g})$ given by the
following relations [17]

(5) $\displaystyle U_{0}(\mathfrak{g})=\mathbb{F},\quad
U_{h}(\mathfrak{g})U_{k}(\mathfrak{g})\subset U_{h+k}(\mathfrak{g}),\quad
U_{h}(\mathfrak{g})\subset U_{h+k}(\mathfrak{g}),\quad\forall k\geq 1.$

Clearly, for each $k\geq 1$ we can define the grading blocks

(6) $U^{k}(\mathfrak{g})=U_{k}(\mathfrak{g})/U_{k-1}(\mathfrak{g}),$

from which it follows that $U(\mathfrak{g})\simeq\bigoplus_{k\geq
0}U^{k}(\mathfrak{g})$.

We now focus our attention on the (noncommutative) Poisson centralizers in
$\left(S(\mathfrak{g}),\\{\cdot,\cdot\\}_{o}\right)$ (for similar
constructions, see [2, 16, 25] and references therein). Let
$S^{k}(\mathfrak{g})$ be the subspace of $S(\mathfrak{g})$ formed by
homogeneous polynomials of degree $k$. Clearly this induces the decomposition

(7) $\displaystyle S(\mathfrak{g})=\bigoplus_{k\geq 0}S^{k}(\mathfrak{g}).$

It follows that, for any $p\in S(\mathfrak{g})$, the polynomial decomposes as
$p=\sum_{k\geq 0}p_{k}$, where $p_{k}\in S^{k}(\mathfrak{g})$ for all $k\geq
0$. We define the vector space of $\mathfrak{a}$-invariant $k$-homogeneous
polynomials as

$\displaystyle S^{k}(\mathfrak{g})^{\mathfrak{a}}=\left\\{p_{k}\in
S^{k}(\mathfrak{g}):\\{x,p_{k}\\}_{o}=0\quad\forall
x\in\mathfrak{a}^{*}\right\\},$

where $p_{k}(x_{1},\ldots,x_{n})$ is a homogeneous polynomial of degree $k\geq
0$ with the generic form

(8) $\displaystyle
p_{k}(x_{1},\ldots,x_{n})=\sum_{i_{1}+\cdots+i_{n}=k}\Gamma_{i_{1},\ldots,i_{n}}\,x_{1}^{i_{1}}\cdots
x_{n}^{i_{n}},\quad\Gamma_{i_{1},\ldots,i_{n}}\in\mathbb{F}.$

By definition, in order to find a suitable basis for centralizer subalgebras,
all $\mathfrak{a}$-invariant (linearly independent) indecomposable homogeneous
polynomial solutions of the system of partial differential equations

(9)
$\displaystyle\bar{X}_{j}(p_{k})(x_{1},\ldots,x_{n})=\\{x_{j},p_{k}\\}_{o}=\sum_{1\leq
l,i\leq n}C_{ji}^{l}x_{l}\dfrac{\partial p_{k}}{\partial x_{i}}=0,\text{
}\quad 1\leq j\leq\dim\mathfrak{a}=s$

must be found.

We observe that, for an arbitrary (finite-dimensional) Lie algebra, a linear
basis in $S(\mathfrak{g})^{\mathfrak{a}}$ is not necessarily canonical. In
other words, it cannot be ensured that the associated Poisson algebra is
finitely-generated. However, for the case of semisimple Lie algebras
$\mathfrak{g}$, it can be shown that $S(\mathfrak{g})$ is Noetherian, from
which we can conclude that the centralizer $S(\mathfrak{g})^{\mathfrak{a}}$ is
finitely-generated [17, Chapter 2]. Hence, we always assume that the system
$\eqref{eq:func}$ admits an integrity basis formed by polynomials. This
implies that, once a maximal set of indecomposable polynomials
$\left\\{p_{k_{1}},\dots,p_{k_{s}}\right\\}$ has been found, there always
exists some integer $g_{j}\in\mathbb{N}$ such that $p_{k_{j}+g_{j}}$ is
decomposable for all $k\geq 1$. By decomposability of a polynomial $p\in
S(\mathfrak{g})$ we mean that there exists some polynomial $p^{\prime}\in
S(\mathfrak{g})$ of lesser degree such that $p\equiv 0\mod p^{\prime}$, i.e.,
$p^{\prime}$ divides $p$. We also observe that the linear independence of the
basis elements (in the centralizer of a subalgebra) does not generally imply
the algebraic independence. As a matter of fact, the maximal number of
functionally independent solutions of $\eqref{eq:func}$ is known to be given
by [2, 30]

$\displaystyle\mathcal{N}(\mathfrak{g})=\dim\mathfrak{g}-\mathrm{rank}\left(C_{ji}^{l}x_{l}\right),\quad
1\leq j\leq s,\text{ }1\leq k,i\leq\dim\,\mathfrak{g},$

where $\left(C_{ji}^{l}x_{l}\right)$ denotes the matrix associated to the
commutator table of the Lie algebra $\mathfrak{g}$.

Let $\mathfrak{a}$ be Lie subalgebra of $\mathfrak{g}$. In the following, we
propose an algorithm for the construction of indecomposable generators
(linearly independent monomials) in $S(\mathfrak{g})^{\mathfrak{a}}$. This
procedure will allow us to build a basis for
${S^{k}(\mathfrak{g})}^{\mathfrak{a}}$ systematically. Starting from degree
one, it is clear that ${S^{1}(\mathfrak{g})}^{\mathfrak{a}}=\mathfrak{a}$ if
$\mathfrak{a}$ is a maximal Abelian subalgebra, otherwise
${S^{1}(\mathfrak{g})}^{\mathfrak{a}}$ coincides with the centralizer of
$\mathfrak{a}$ in $\mathfrak{g}$. Without loss of generality, suppose that all
indecomposable degree-one monomials are given by

$\textbf{q}_{1}=\left\\{p_{1,1},\ldots,p_{1,m_{1}}\right\\}.$

We now proceed with the construction of indecomposable quadratic solutions of
$\eqref{eq:func}$. It is clear that for any
$p_{1},p_{1}^{\prime}\in\textbf{q}_{1}$, the product
$p_{2}=p_{1}p_{1}^{\prime}$ is decomposable, hence any quadratic solutions
that are the product of elements in $\textbf{q}_{1}$ must be discarded. It
follows that (indecomposable) quadratic homogeneous polynomials belonging to
the centralizer must depend on generators of $\mathfrak{g}$ that do not belong
to the subalgebra. Thus let

$\displaystyle\textbf{q}_{2}=\left\\{p_{2,1},\ldots,p_{2,m_{2}}\right\\}.$

Iterating the procedure, and discarding all elements that can be expressed as
products of lower-order polynomials, all indecomposable homogeneous
polynomials up to a certain degree $\zeta$ that are solutions to
$\eqref{eq:func}$ can be deduced:

(10)
$\displaystyle\textbf{Q}_{\zeta}=\bigcup_{h=1}^{\zeta}\textbf{q}_{h},\quad\textbf{q}_{h}=\\{p_{h,1},\ldots,p_{h,m_{h}}\\},\;1\leq
h\leq\zeta.$

In the following, we shall assume that $\zeta$ denotes the maximum degree of
an indecomposable homogeneous polynomial in the set of all commutants
$\textbf{Q}_{\zeta}$, i.e., that any homogeneous polynomial of degree greater
than $\zeta$ is decomposable, and hence not contained in $\textbf{Q}_{\zeta}.$
It is clear that
$\mathcal{Q}_{h}:=\mathbb{F}\langle\textbf{q}_{h}\rangle\subset
S^{h}(\mathfrak{g})^{\mathfrak{a}}$ forms a vector space for each $h$, and
thus $\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle$ is also a vector space,
where $\mathbb{F}$ is an arbitrary field. It is straightforward to verify that
we have the filtration

$\mathcal{Q}_{1}\subset\mathbb{F}\langle\textbf{Q}_{1}\rangle\subset\dots\subset\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle,\quad\dim_{FL}\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle=m_{1}+\cdots+m_{\zeta}.$

Notice that $\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle$ is infinite-
dimensional. Here $\dim_{FL}$ denotes the number of indecomposable monomials
that generate $\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle.$ Now, for any
$p_{h}\in\textbf{q}_{h}$, $p_{\ell}\in\textbf{q}_{\ell}$ and
$x_{j}\in\mathfrak{a}$, the Jacobi identity for the Poisson–Lie bracket (1)
gives

(11)
$\\{x_{j},\\{p_{h},p_{\ell}\\}_{o}\\}_{o}+\\{p_{\ell},\\{x_{j},p_{h}\\}_{o}\\}_{o}+\\{p_{h},\\{p_{\ell},x_{j}\\}_{o}\\}_{o}=0,$

hence implying that $\\{p_{h},p_{\ell}\\}_{o}$ is a solution of (9). As
$\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle$ encompasses all indecomposable
linearly independent solutions of this system, there exist constants
$\Gamma_{r_{1},\ldots,r_{k}}^{m_{r_{1}},\ldots,m_{r_{k}}}\in\mathbb{F}$ such
that

(12)
$\\{p_{h},p_{\ell}\\}_{o}=\sum_{r_{1}+\cdots+r_{k}=\ell+h-1}\Gamma_{r_{1},\ldots,r_{k}}^{m_{r_{1}},\ldots,m_{r_{k}}}p_{r_{1},m_{r_{1}}}\cdots
p_{r_{k},m_{r_{k}}},$

where $1\leq r_{i}\leq\zeta$ for $1\leq i\leq k$. It is a routine computation
to verify that the Leibniz rule is satisfied. This operation endows the set of
polynomial solutions of (1) with a Poisson structure (that will be denoted by
$\\{\cdot,\cdot\\}$, instead of $\\{\cdot,\cdot\\}_{o}$), where
$\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle$ can be seen as a system of
generators. We denote the polynomial algebra generated by
$\mathbb{F}\langle\textbf{Q}_{\zeta}\rangle$ by
$\left(\mathcal{Q}_{\mathfrak{g}}(d),\\{\cdot,\cdot\\}\right)$, where the
degree $d$ is defined as the maximal number of elements intervening in the
decomposition (12) for $p_{h},p_{\ell}\in\textbf{Q}_{\zeta}$. i.e., $d$ is
given by

(13) $\displaystyle
d:=\max_{h_{1}+\cdots+h_{j}\leq\ell+h-1}\mathcal{N}\left(p_{h_{1},m_{h_{1}}}\cdots
p_{h_{j},m_{h_{j}}}\right),$

where $\mathcal{N}\left(p_{h_{1},m_{h_{1}}}\cdots p_{h_{j},m_{h_{j}}}\right)$
is the number of $p_{h,m_{h}}$ appearing in $p_{h_{1},m_{h_{1}}}\cdots
p_{h_{j},m_{h_{j}}}.$ In particular, if $d=0$, then
$\mathcal{Q}_{\mathfrak{g}}(0)$ is an Abelian algebra.

We now consider the setting in the enveloping algebra $U(\mathfrak{g})$ (it
can be referred to as the quantized setting). Recall that there is a well-
defined canonical isomorphism (see [1] for details)

$\displaystyle\Lambda:S(\mathfrak{g})$ $\displaystyle\rightarrow
U(\mathfrak{g}),$ (14) $\displaystyle p\left(x_{1},\ldots,x_{n}\right)$
$\displaystyle\mapsto\Lambda\left(p\left(x_{1},\ldots,x_{n}\right)\right)=P\left(X_{1},\ldots,X_{n}\right),$

in terms of the monomial basis of $S(\mathfrak{g})$ given by

(15) $\displaystyle\Lambda(x_{i_{1}}\cdots
x_{i_{n}})=\frac{1}{n!}\sum_{\sigma\in S_{n}}X_{i_{\sigma(1)}}\cdots
X_{i_{\sigma(n)}}$

with $S_{n}$ being the permutation group on the set $\\{1,\ldots,n\\}$, such
that $\Lambda\big{(}\tilde{X}(p(x_{1},\ldots,x_{n})\big{)}=[X,\Lambda(p)]_{o}$
for any $X\in\mathfrak{g}$. In particular, for any $p\in S^{k}(\mathfrak{g})$
and $q\in S^{\ell}(\mathfrak{g}),$ we have

$\displaystyle\deg\left(\Lambda(pq)\right)=k+\ell,\qquad\Lambda(pq)-\Lambda(p)\Lambda(q)\in
U_{k+\ell-1}(\mathfrak{g}).$

It follows that $\Lambda(p_{g}(x_{1},\ldots,x_{n}))=P_{g}(X_{1},\ldots,X_{n})$
will be non-commutative, functionally independent polynomials in
$U(\mathfrak{g}),$ where $\zeta$ is the maximum degree of the generators.
Moreover, it is an isomorphism of filtered algebras. That is,
$\Lambda_{k}\left(S^{k}(\mathfrak{g})\right)=U^{k}(\mathfrak{g})$, where
$\Lambda_{k}=\Lambda|_{S^{k}(\mathfrak{g})}$ and
$U^{k}(\mathfrak{g})=\mathrm{span}\left\\{X_{1}^{i_{1}}\cdots
X_{n}^{i_{n}}:i_{1}+\cdots+i_{n}=k\right\\}$ (see (6)). From the Poincaré-
Birkhoff-Witt theorem (PBW in short) it can be easily deduced that the
dimension of each filtration block is

(16) $\displaystyle\dim
U^{k}(\mathfrak{g})=\dim\frac{U_{k}(\mathfrak{g})}{U_{k-1}(\mathfrak{g})}=\binom{\dim\mathfrak{g}+k-1}{k}.$

Define

$\displaystyle U^{k}(\mathfrak{g})^{\mathfrak{a}}$ $\displaystyle=\left\\{Y\in
U^{k}(\mathfrak{g}):[X,Y]_{o}=0\quad\forall X\in\mathfrak{a}\right\\}.$

By construction, it follows that
$U(\mathfrak{g})^{\mathfrak{a}}=\bigoplus_{k\geq
0}U^{k}(\mathfrak{g})^{\mathfrak{a}}$ and
$S(\mathfrak{g})^{\mathfrak{a}}=\bigoplus_{k\geq
0}S^{k}(\mathfrak{g})^{\mathfrak{a}}.$ In particular, $\Lambda_{k}$ induces an
algebra isomorphism between $U(\mathfrak{g})^{\mathfrak{a}}$ and
$S(\mathfrak{g})^{\mathfrak{a}}$. Let
$\hat{\textbf{q}}_{k}=\left\\{P_{k,1},\ldots,P_{k,m_{k}}\right\\}=\Lambda\big{(}\textbf{q}_{h}\big{)}$
be the set of all indecomposable homogeneous representatives of degree $k$,
that is, $P_{k,j}=\Lambda(p_{k,j})$ for all $1\leq j\leq m_{k}$, and let
$\zeta$ be the maximal degree of homogeneous representatives. Define the set
of indecomposable polynomials by

$\tilde{\textbf{Q}}_{\zeta}=\bigcup_{k=1}^{\zeta}\tilde{\textbf{q}}_{k}.$

As before, we can consider the linear space spanned by
$\tilde{\textbf{Q}}_{\zeta}$ with dimension $m_{1}+\cdots+m_{\zeta}$. In
analogy to the commutative case, for any $P_{k}\in\tilde{\textbf{q}}_{k}$ and
$P_{\ell}\in\tilde{\textbf{q}}_{\ell}$, we define the commutator by

(17)
$\displaystyle[P_{k},P_{\ell}]=\sum_{h_{1}+\cdots+h_{j}=\ell+k-1}\Gamma_{h_{1},\ldots,h_{j}}^{m_{h_{1}},\ldots,m_{h_{j}}}P_{h_{1},m_{h_{1}}}\cdots
P_{h_{j},m_{h_{j}}},$

where $\Gamma_{h_{1},\ldots,h_{j}}^{m_{h_{1}},\ldots,m_{h_{j}}}$ are
constants. We denote by $\tilde{\mathcal{Q}}_{\mathfrak{g}}(d)$ the algebra
generated by $\tilde{\textbf{Q}}_{g}$, where $d$ is the degree of
$\tilde{\mathcal{Q}}_{\mathfrak{g}}(d)$. We conclude that
$\left(\tilde{\mathcal{Q}}_{\mathfrak{g}}(d),[\cdot,\cdot]\right)$ forms an
associative algebra in $U(\mathfrak{g})$. From these constructions, we can
define algebraic Hamiltonians and their corresponding superintegrable systems
in $S\left(\mathfrak{g}\right)$, which can be considered as classical
superintegrable systems. Then, applying the symmetrization mapping $\Lambda$,
we can deduce a quantized version of these systems.

######  Definition 2.3.

[4]. Let $\mathfrak{a}^{*}\subset\mathfrak{g}^{*}$ be a Lie subalgebra with an
ordered basis $\beta_{\mathfrak{a}^{*}}=\\{x_{1},\ldots,x_{s}\\}$, and let
$\left(\mathcal{Q}_{\mathfrak{g}}(d),\\{\cdot,\cdot\\}\right)$ be a polynomial
algebra in $S(\mathfrak{g})^{\mathfrak{a}}$. An algebraic Hamiltonian with
respect to $\mathcal{Q}_{\mathfrak{g}}(d)$ is given by

(18)
$\displaystyle\mathcal{H}=\sum_{i_{1},\ldots,i_{k}}^{s}\gamma_{i_{1},\ldots,i_{k}}x_{i_{1}}\cdots
x_{i_{k}}+\sum_{t}\gamma_{t}\mathcal{C}_{t}\in\mathcal{Z}\left(\mathcal{Q}_{\mathfrak{g}}(d)\right),$

where $\gamma_{i_{1},\ldots,i_{k}}\in\mathbb{F}$, $\mathcal{C}_{t}$ are
Casimir invariants of $\mathfrak{g}^{*}$ and
$\mathcal{Z}\left(\mathcal{Q}_{\mathfrak{g}}(d)\right)$ is the center of
$\mathcal{Q}_{\mathfrak{g}}(d).$

Depending on the structure of the subalgebra $\mathfrak{a}$,
$\mathcal{Q}_{\mathfrak{g}}(d)$ may contain a sufficiently high number of
algebraic independent polynomials $\\{p_{1},\dots,p_{r}\\}$ such that the set
$S=\\{\mathcal{H}\\}\cup\\{p_{1},\dots,p_{r}\\}$ forms a superintegrable
system, where the maximal number $r$ of elements commuting with the algebraic
Hamiltonian $\mathcal{H}$ is upper-bounded by

$r\leq\dim\mathfrak{g}-\dim\mathfrak{a}-1.$

######  Remark 2.4.

Using the symmetrization map $\Lambda$, for any $p\in
S(\mathfrak{g})^{\mathfrak{a}}$, we have $P=\Lambda(p)\in U(\mathfrak{g})$
such that $[\tilde{\mathcal{H}},P]=\Lambda(\\{\mathcal{H},p\\})=0,$ where
$\tilde{H}$ is the Hamiltonian, with respect to the ordered basis of
$\mathfrak{g}$, adopts of the form

(19)
$\displaystyle\tilde{\mathcal{H}}=\sum_{i_{1},\ldots,i_{k}}^{s}\Gamma_{i_{1},\ldots,i_{k}}X_{i_{1}}\ldots
X_{i_{k}}+\sum_{t}\sigma_{t}C_{t}+{\rm L.O.T.},$

where $\Gamma_{i_{1},\ldots,i_{k}},\sigma_{t}\in\mathbb{F}$, $C_{t}$ denotes
the Casimir operators of $\mathfrak{g}$ and L.O.T. refers to the lower-order
terms arising from expressing $\Lambda(\mathcal{H})$ in the ordered basis of
$U(\mathfrak{g})$. The number of functionally independent integrals of the
motion associated to $\tilde{\mathcal{H}}$ can be determined using
$\eqref{eq:func}$. In the physical context, as Hamiltonians are usually
identified with quadratic differential operators connected to a Schrödinger
equation, $\tilde{\mathcal{H}}$ needs to be a quadratic differential operator.
To this extent, suitable realizations of the generators $X_{i}$ can be chosen,
such that first summation in $\tilde{\mathcal{H}}$ does not produce
differential operators with order higher than 2 [8].

##  3\. Construction of Cartan centralizers of semisimple Lie algebras

In the following, we assume that $\mathfrak{g}$ is a rank $n$ complex
semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Hence,
$\mathfrak{g}$ is isomorphic to $A_{n}\cong\mathfrak{sl}(n+1,\mathbb{C})$,
$B_{n}\cong\mathfrak{so}(2n+1,\mathbb{C})$,
$C_{n}\cong\mathfrak{sp}(2n,\mathbb{C})$,
$D_{n}\cong\mathfrak{so}(2n,\mathbb{C})$ or one of the exceptional Lie
algebras $G_{2}$, $F_{4},$ $E_{6},E_{7}$ or $E_{8}$ [23]. In this section, we
will provide the classification of the elements in the Cartan commutants of
all simple classical Lie algebras on an ordered basis determined by the root
space decomposition. Commutants of the nilradical of Borel subalgebras of
semisimple Lie algebras, which are deeply related to decompositions of
enveloping algebras, have been considered in [10].

Let $\Phi$ denote the root system system associated to the complex semisimple
algebra $\mathfrak{g}$ and let $\Delta=\\{\beta_{1},\ldots,\beta_{n}\\}$
denote a basis of simple roots of $\Phi$. With respect to the root system
$\Phi$, the Lie algebra $\mathfrak{g}$ admits the decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{g}_{+}\oplus\mathfrak{g}_{-},$ where
$\mathfrak{g}_{\pm}=\bigoplus_{\beta\in\Phi^{\pm}}\mathfrak{g}_{\beta}$ and
$\mathfrak{g}_{\beta}$ is a one dimensional subalgebra spanned by a root
vector. Let $\mathfrak{h}=\mathrm{span}\\{H_{1},\ldots,H_{n}\\}$,
$\mathfrak{g}_{+}:=\mathrm{span}\\{E_{1},\ldots,E_{|\Phi^{+}|}\\}$. By Serre-
Chevalley relations, we have

$\displaystyle[H_{i},H_{j}]_{o}=$ $\displaystyle 0,\text{
}[E_{i},E_{j}]_{o}=\delta_{ij}H_{i},\text{
}[H_{i},E_{j}]_{o}=a_{ij}E_{j},\text{
}[H_{i},\hat{E}_{j}]_{o}=-a_{ij}\hat{E}_{j},$ (20)
$\displaystyle\left(\mathrm{ad}E_{i}\right)^{-a_{ij}+1}E_{j}=\left(\mathrm{ad}\hat{E}_{i}\right)^{-a_{ij}+1}\hat{E}_{j}=0$

for all $1\leq i\neq j\leq n,$ where $\hat{E}_{j}$ are root vectors
corresponding to negative roots.

In what follows, we denote the Cartan commutant by
$U(\mathfrak{g})^{\mathfrak{h}}.$ Notice that $U(\mathfrak{g})\cong
U(\mathfrak{h})\otimes U^{+}\otimes U^{-}$ is a vector space isomorphism,
where

$U^{+}=U(\mathfrak{g}_{+})=\bigotimes_{\beta\in\Phi^{+}}U\left(\mathfrak{g}_{\beta}\right)\text{
and
}U^{-}=U(\mathfrak{g}_{-})=\bigotimes_{\beta\in\Phi^{+}}U\left(\mathfrak{g}_{-\beta}\right).$

The subalgebras $U^{+}$ and $U^{-}$ are generated by $E_{\gamma_{j}}$ and
$\hat{E}_{w_{j}(\gamma_{j})}$ denoted by $E_{j}$ and $\hat{E}_{w_{j}}$
respectively, where $w_{j}$ is an element of the Weyl group such that
$w_{j}(\gamma_{j})=-\gamma_{j}.$ By the PBW-theorem, we can write

(21) $\displaystyle U(\mathfrak{g})=\mathrm{span}\left\\{H_{1}^{k_{1}}\cdots
H_{n}^{k_{n}}E_{1}^{l_{1}}\cdots
E_{t}^{l_{t}}\hat{E}_{w_{1}}^{m_{1}}\cdots\hat{E}_{w_{t}}^{m_{t}}:l_{t},m_{t},k_{j}\in\mathbb{N}_{0}\right\\}.$

Let
$\textbf{X}=\left(H_{1},\ldots,H_{n},E_{1},\ldots,E_{|\Phi^{+}|},\hat{E}_{1},\ldots,\hat{E}_{|\Phi^{-}|}\right)$.
Then for any $P(\textbf{X})\in U(\mathfrak{g}),$ one can write

$\displaystyle P(\textbf{X})=$
$\displaystyle\sum_{\sum_{j,t}k_{j}+m_{t}+l_{t}\leq
i}\Gamma_{l_{1},\ldots,l_{t},m_{1},\ldots,m_{t}}^{k_{1},\ldots,k_{n}}H_{1}^{k_{1}}\cdots
H_{n}^{k_{n}}E_{1}^{l_{1}}\cdots
E_{t}^{l_{t}}\hat{E}_{w_{1}}^{m_{1}}\cdots\hat{E}_{w_{t}}^{m_{t}}$
$\displaystyle=$ $\displaystyle\sum_{\sum_{j,t}k_{j}+m_{t}+l_{t}\leq
i}\Gamma_{\textbf{l},\textbf{m}}^{\textbf{k}}\textbf{H}^{\textbf{k}}\textbf{E}^{\textbf{l}}\hat{\textbf{E}}^{\textbf{m}},$

where $\textbf{l}=(l_{1},\ldots,l_{t}),$
$\textbf{m}=(m_{1},\ldots,m_{t})\in\mathbb{N}_{0}^{t}$ and
$\textbf{k}=(k_{1},\ldots,k_{n})\in\mathbb{N}_{0}^{n}$,
$\Gamma_{\textbf{l},\textbf{m}}^{\textbf{k}}$ are constants and H, E and
$\hat{\textbf{E}}$ represents the multiplication of the Cartan elements and
root vectors.

Denote the dual space of $\mathfrak{g}$ by $\mathfrak{g}^{*}$, and let
$\textbf{x}=\big{(}h_{1},\ldots,h_{n},\epsilon_{1},\ldots,\epsilon_{|\Phi^{+}|},\hat{\epsilon}_{1},\ldots,\hat{\epsilon}_{|\Phi^{-}|}\big{)}$
denote the coordinates of a dual basis. As $\mathfrak{g}$ is semisimple, we
can then identify $\mathfrak{g}$ with $\mathfrak{g}^{*}$ using the Killing
form $\kappa$. We proceed with our study to the symmetric algebra
$S(\mathfrak{g})$. By definition, the kernel of the coadjoint action of
$\mathfrak{h}$ on $S(\mathfrak{g})$ is given by

(22) $\displaystyle S(\mathfrak{g})^{\mathfrak{h}}=\left\\{p\in
S(\mathfrak{g}):\text{ }\\{p,h\\}_{o}=0\quad\forall
h\in\mathfrak{h}^{*}\right\\}.$

It follows that $S(\mathfrak{g})\cong S(\mathfrak{h})\otimes S(\mathfrak{u})$
is a vector space isomorphism, where
$\mathfrak{u}=\mathfrak{g}_{+}\oplus\mathfrak{g}_{-}.$ Since $S(\mathfrak{h})$
is commutative, in the analytical frame, the PDE system corresponding to
$\eqref{eq:PDE20}$ becomes

(23)
$\displaystyle\bar{H}_{j}(p)(\textbf{x})=\sum_{\beta\in\Phi^{+}}\beta(h_{j})\epsilon_{\beta}\dfrac{\partial}{\partial\epsilon_{\beta}}-\sum_{\beta\in\Phi^{+}}\beta(h_{j})\epsilon_{-\beta}\dfrac{\partial}{\partial\epsilon_{-\beta}}=0,$

where $\beta(h_{j})$ is the weight of the roots $\beta$ with respect to the
Cartan generators. Our aim is to determine the (homogeneous) polynomial
solutions of the system (23) of degree $i\geq 1$

(24) $\displaystyle
p_{i}(\textbf{x})=\sum_{\sum_{j,t}k_{j}+l_{t}+m_{t}=i}\Gamma_{\textbf{m},\textbf{l}}^{\textbf{k}}\textbf{h}^{\textbf{k}}\textbf{e}^{\textbf{l}}\hat{\textbf{e}}^{\textbf{m}}\in
S(\mathfrak{g}).$

Here $\textbf{e}=\epsilon_{\beta_{1}}\cdots\epsilon_{\beta_{t}}$ and
$\hat{\textbf{e}}=\epsilon_{-\beta_{1}}\cdots\epsilon_{-\beta_{t}},$
$\textbf{k},\textbf{l}$ and m are integer tuples defined above and
$\Gamma_{\textbf{m},\textbf{l}}^{\textbf{k}}$ are constants. Using
$\eqref{eq:muli}$ and the structure tensor of $\mathfrak{g}^{*}$, the identity
$\eqref{eq:PDEs}$ can be reformulated as

(25)
$\displaystyle\bar{H}_{j}(p)(\textbf{x})=\sum_{s=2}^{n}F_{s}(h_{1},\ldots,h_{n+1})\Gamma_{\textbf{k},\textbf{m}}\left(\left(l_{1}-m_{1}\right)a_{1j}+\cdots+\left(l_{t}-m_{t}\right)a_{tj}\right)\textbf{e}^{\textbf{l}}\hat{\textbf{e}}^{\textbf{m}},$

where $a_{ij}$ are the structure constants from Serre-Chevalley relations
$\eqref{eq:18}$. Using $\eqref{eq:muli}$ and the preceding expression, a
monomial $\epsilon_{i_{1}}\ldots\epsilon_{i_{r}}\in S(\mathfrak{u})$ is a
solution of $\eqref{eq:PDEs}$ if and only if

(26) $\displaystyle\epsilon_{i_{1}}(h_{j})+\cdots+\epsilon_{i_{r}}(h_{j})=0.$

That is, the weight of the monomial with respect to each generator of the
Cartan subalgebras must be zero. We now consider a setting that exploits root
systems instead of weights. In this context, we observe that $p\in
U(\mathfrak{g})^{\mathfrak{h}}$ is linearly independent and indecomposable if
and only if there exists a root $\beta_{k}$ such that
$\mathrm{lgh}(\gamma_{k})=\max_{1\leq j\leq
r}\left\\{\mathrm{lgh}(\gamma_{j})\right\\}$ and $\sum_{j}^{r}\gamma_{j}=0$
with $\gamma_{j}=\sum_{i}^{j}\lambda_{i}\beta_{i}\in\Phi$ for all $j$, where
$\mathrm{lgh}:\Phi\rightarrow\mathbb{N}_{0}$ is the length of a root in
$\Phi.$ Moreover, $\deg p=\mathrm{lgh}(\beta_{k})+1.$

######  Definition 3.1.

Let $\mathfrak{g}$ and $U(\mathfrak{g})^{\mathfrak{h}}$ be the same as above.
For any $p\in U(\mathfrak{g})^{\mathfrak{h}}$, we call a degree polynomial $h$
$p$ to be decomposable if there exists some $p_{j}^{\prime}\in
U(\mathfrak{g})^{\mathfrak{h}}$ such that $p=\prod_{j\in J}p_{j}^{\prime}$
with an index set $|J|<\infty$ and
$R\left(p_{j}^{\prime}\right)=\sum_{i}^{j}\lambda_{i}\beta_{i}=0.$

Following the algorithm in Section $\ref{2},$ we will construct a polynomial
algebra under a bilinear form $\\{\cdot,\cdot\\}$ defined in
$\eqref{eq:newpoi}$. Let $\textbf{q}_{h}$ be the set that contains all the
monomials of degree $h$ that satisfy $\\{h_{i},p_{h}\\}_{o}=0.$ We now
construct the elements in each layer $\mathcal{Q}_{h}$ and show that
$\mathrm{span}\,\textbf{q}_{h}=\mathcal{Q}_{h}$ defines a vector space. Since
$\deg p=2$, for any $\beta\in\Phi^{+},$ it is clear that
$\epsilon_{\beta}\hat{\epsilon}_{\beta}$ are degree $2$ monomials, which are
contained in the set $\textbf{q}_{2},$ where
$\hat{\epsilon}_{\beta}:=\epsilon_{-\beta}.$ Moreover, we observe that
$\mathrm{span}\,\textbf{q}_{2}=\mathcal{Q}_{2}$ and
$\dim\mathcal{Q}_{2}=\left|\Phi^{+}\right|$. If $p\in\textbf{q}_{3},$ it is
clear that the decomposable monomials of the form
$h_{i}\epsilon_{\beta}\hat{\epsilon}_{\beta}$ need to be discarded for any
$\beta\in\Phi^{+}$ and $1\leq i\leq\mathrm{rank}\mathfrak{g}$. Hence, for any
$\gamma_{1},\gamma_{2}\in\Phi,$ an indecomposable degree $3$ monomial must
have the form of
$\epsilon_{\gamma_{1}}\epsilon_{\gamma_{2}}\hat{\epsilon}_{\gamma_{1}+\gamma_{2}}.$
By recursion, for any $p\in\textbf{q}_{h}$, there exists a maximal length root
$\epsilon_{\gamma_{j}}$ with $\gamma_{j}=\sum_{i}^{j-1}c_{i}\gamma_{i}$ such
that $p$ is the form of
$\epsilon_{\gamma_{1}}\cdots\epsilon_{\gamma_{j-1}}\hat{\epsilon}_{\gamma_{j}}$
where $\gamma_{i}\in\Phi$ for all $i.$ It is thus sufficient to express
$\mathcal{Q}_{h}$ as

(27)
$\displaystyle\mathcal{Q}_{h}=\mathrm{span}\,\textbf{q}_{h}=\left\\{\epsilon_{\gamma_{1}}^{l_{1}}\cdots\epsilon_{\gamma_{j}}^{l_{j}}:\text{
}\sum_{i}^{j}l_{i}\gamma_{i}=0,\text{ }\sum_{i}^{j}l_{i}=h\right\\}$

for all $h\geq 2.$ Note that for each generator
$\epsilon_{\gamma_{1}}\cdots\epsilon_{-\gamma_{1}-\cdots-\gamma_{r-1}}\in\mathcal{Q}_{h}$
with $h\geq 2$, the following numerical relation holds

$R\left(\epsilon_{\gamma_{1}}\cdots\epsilon_{-\gamma_{1}-\cdots-\gamma_{r-1}}\right)=\gamma_{1}+\gamma_{2}+\cdots+\left(-\gamma_{1}-\cdots-\gamma_{r-1}\right)=0.$

It follows that the composition of sets

(28) $\displaystyle\textbf{Q}_{\zeta}=\left\\{h_{i}\right\\}\cup\bigcup_{h\geq
2}^{\zeta}\textbf{q}_{h}=\left\\{\epsilon_{\gamma}\epsilon_{-\gamma},\epsilon_{\gamma_{1}}\epsilon_{\gamma_{2}}\epsilon_{-\gamma_{1}-\gamma_{2}},\ldots,\epsilon_{\gamma_{1}}\cdots\epsilon_{-\gamma_{1}-\cdots-\gamma_{r-1}}\right\\}$

gives rise to a basis. Let $\mathbb{C}\langle\textbf{Q}_{\zeta}\rangle$ denote
the vector space consisting of all linearly independent homogeneous polynomial
solutions of $\eqref{eq:PDEs}$. Using (4) and (12), the bilinear product

(29)
$\displaystyle\left\\{\epsilon_{\gamma_{1}}\cdots\epsilon_{\gamma_{t}}\hat{\epsilon}_{\gamma_{t+1}},\epsilon_{\theta_{1}}\cdots\epsilon_{\theta_{s}}\hat{\epsilon}_{\theta_{s+1}}\right\\}=\sum_{k,l}^{t,s}\left\\{\epsilon_{\gamma_{k}},\epsilon_{\theta_{l}}\right\\}_{o}\epsilon_{\gamma_{i}}\prod_{i\neq
k,j\neq
l}\epsilon_{\theta_{j}}\hat{\epsilon}_{\theta_{s+1}}\hat{\epsilon}_{\gamma_{t+1}}+\ldots$

is well defined and can be expressed as sums of products of elements in
$\mathbb{C}\langle\textbf{Q}_{\zeta}\rangle$, satisfying the constraint
$R\left(\prod_{i\neq k,j\neq
l}\\{\epsilon_{\gamma_{k}},\epsilon_{\theta_{l}}\\}_{o}\epsilon_{\gamma_{i}}\epsilon_{\theta_{j}}\hat{\epsilon}_{\theta_{s+1}}\hat{\epsilon}_{\gamma_{t+1}}\right)=0$,
showing that the latter generates a polynomial algebra. As shown previously,
it actually inherits a Poisson structure from (1). We denote by
$\mathcal{Q}_{\mathfrak{g}}(d)$ the polynomial algebra generated by
$\mathbb{C}\langle\textbf{Q}_{\zeta}\rangle$, where $d$ is the maximal number
of elements intervening in the product (29).

Let $\hat{\cdot}$ denote the operator that transforms a positive root vector
into its negative root vector. That is, for any $\beta\in\Phi^{+}$ and
$E_{\beta}\in\mathfrak{g}_{+}$, we have that
$\hat{E}_{\beta}=E_{-\beta}\in\mathfrak{g}_{-\beta}.$ It is clear that if
$p\in\mathcal{Q}_{\mathfrak{g}}(d),$ then
$\hat{p}(\textbf{x}):=p(\hat{\textbf{x}})\in\mathcal{Q}_{\mathfrak{g}}(d)$,
were x is a commuting coordinate of $\mathfrak{g}^{*}$.

By construction, we have the linear decomposition
$\mathcal{Q}_{\mathfrak{g}}(d)=\mathfrak{h}\oplus\bigoplus_{h\geq
2}^{\zeta}\mathcal{Q}_{h}$ with respect to a $\mathcal{W}$-invariant inner
product. In particular, given a chain of Lie algebras
$\mathfrak{g}_{0}\subset\mathfrak{g}_{1}\subset\ldots\subset\mathfrak{g}_{t}$,
their corresponding polynomial algebras induce the following subspace chain
$\mathcal{Q}_{\mathfrak{g}_{1}}(d_{1})\subset\mathcal{Q}_{\mathfrak{g}_{2}}(d_{2})\subset\ldots\subset\mathcal{Q}_{\mathfrak{g}_{t}}(d_{t})$
with $d_{1}\leq d_{2}\leq\ldots\leq d_{t}\in\mathbb{N}$. For each root
$\beta\in\Phi,$ the Weyl group of a Lie algebra can be interpreted as the
reflection of $\beta$ along the orthogonal hyperplane to a root $\alpha$ given
by

(30) $\displaystyle
s_{\alpha}(\beta)=\beta-2\frac{\langle\alpha,\beta\rangle}{\langle\alpha,\alpha\rangle}\alpha=\beta-\beta(H_{\alpha})\alpha.$

Let $\mathcal{P}(\Phi)$ denote the root polytope corresponding to $\Phi$. It
can be shown, from [5], that $R(p)$ is a $\mathcal{W}$-invariant quantity for
any $p\in\mathcal{Q}_{\mathfrak{g}}(d).$ This provides us with another
interpretation of the generators in $\mathcal{Q}_{\mathfrak{g}}(d)$. Without
loss of generality, we can assume that there exist simple reflections
$s_{k}\in\mathcal{W}$ such that
$\gamma_{i}=\prod_{k}^{i}s_{l_{k}}(\gamma_{1}).$ Then

(31)
$\displaystyle\sum_{i}^{j}l_{i}\gamma_{i}=\sum_{i}^{j}l_{i}\prod_{k}^{i}s_{l_{k}}(\gamma_{1})=0.$

Therefore, from $\eqref{eq:31},$ it is sufficient to find a Weyl reflection
function
$w\left(e,s_{i_{1}},\ldots,\prod_{k}^{i}s_{i_{k}}\right)=l_{1}e+\sum_{i_{1},\ldots,i_{k}}\prod_{k}^{i}l_{i_{k}}s_{i_{k}}$
such that $w(e,s_{i_{1}},\ldots,\prod_{k}^{i}s_{i_{k}})$ acts on the root
$\gamma_{1}$ equal to zero, where $e$ is an identity in $\mathcal{W}.$

##  4\. Explicit generators of Polynomial (Poisson) algebras from the $B,C$
and $D$-types of Cartan centralizers

In this section, we will provide the explicit formula of the generators in
Cartan centralizers for semisimple Lie algebras of types $B_{n},C_{n}$ and
$D_{n}$. In what follows, to distinguish different polynomial algebras in
Cartan centralizer of different types, we will denote the polynomial algebras
of type $B_{n},$ $D_{n}$ and $C_{n}$ by
$\mathcal{Q}_{B_{n}}(d),\mathcal{Q}_{D_{n}}(d^{\prime})$ and
$\mathcal{Q}_{C_{n}}(d^{\prime\prime})$ respectively. We also determine the
degree of $\mathcal{Q}_{\mathfrak{g}}(d)$ and the maximum degree of
indecomposable monomials in each of these polynomial algebras. We begin our
analysis with the Cartan centralizer of $B_{n},$ and classify all
indecomposable generators of the polynomial algebra $\mathcal{Q}_{B_{n}}(d)$.
We can then observe that the polynomial algebra $\mathcal{Q}_{B_{n}}(d)$
contains all the Cartan commutant generators of $A_{n}$ and $D_{n}$.
Similarly, the polynomial algebra $\mathcal{Q}_{C_{n}}(d^{\prime\prime})$
contains the commutant generators of $A_{n}$. In other words, we can show that
it admits the following inclusion relations

$\displaystyle\mathcal{Q}_{A_{n-1}}(n-1)\subset\mathcal{Q}_{D_{n}}(d^{\prime})\subset\mathcal{Q}_{B_{n}}(d),\quad\mathcal{Q}_{A_{n-1}}({n-1})\subset\mathcal{Q}_{C_{n}}(d^{\prime\prime}).$

In particular,
$\dim_{FL}\mathcal{Q}_{B_{n}}(d)=\dim_{FL}\mathcal{Q}_{C_{n}}(d^{\prime\prime})$,
where $d,d^{\prime},d^{\prime\prime}\in\mathbb{N}$ is the degree of the
corresponding polynomial algebra. From [6], it is sufficient to deduce the
higher rank Racah algebras contained in $\mathcal{Q}_{B_{n}}(d)$ and
$\mathcal{Q}_{C_{n}}(d^{\prime\prime})$. In order to form a superintegrable
system, we still need to compute the functionally independent generators for
those subalgebras. By definition, the possible Hamiltonian for those
polynomial integrals are given by

(32) $\displaystyle\mathcal{H}=\sum_{i_{1}+\cdots+i_{n}\leq
k}\Gamma_{i_{1},\ldots,i_{n}}h_{1}^{i_{1}}\cdots h_{n}^{i_{n}}\in
S(\mathfrak{h}),$

where $\Gamma_{i_{1},\ldots,i_{n}}$ are some constants. In the following, both
the terminology and the notation for root systems of semisimple Lie algebras
are adapted from [23].

###  4.1. Explicit generators of $U(B_{n})^{\mathfrak{h}}$

We start by introducing some notation of the weights and roots vector from the
semi-simple Lie algebra of type $B_{n}.$ The Cartan subalgebra $\mathfrak{h}$
of $B_{n}$ is formed by elements
$H=\sum_{i=1}^{n}a_{i}\left(E_{i,i}-E_{n+i,n+i}\right),$ where $E_{i,i}$ is
the standard basis in $GL_{2n+1}(\mathbb{R})$. In particular, the basis
elements of $\mathfrak{h}$ are

$\displaystyle H_{i}=E_{i,i}-E_{n+i,n+i}-E_{i+1,i+1}+E_{n+i+1,n+i+1},\quad
2\leq i\leq n.$

The dual space $\mathfrak{so}^{*}(2n+1,\mathbb{C})$ admits the coordinates

$\textbf{x}_{B}=\left(h_{1},\ldots,h_{n},\epsilon_{12}^{-},\ldots,\epsilon_{n-1n}^{-},\right.\left.\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{-},\epsilon_{1},\ldots,\hat{\epsilon}_{n}\right).$

Recall that, in the root system $B_{n}$ in $\mathbb{R}^{n}$, the positive
roots are the $n$ short roots $\alpha_{i}$ and long roots
$\alpha_{i}\pm\alpha_{j}$ with $i\neq j$, where
$\left\\{\alpha_{1},\dots,\alpha_{n}\right\\}$ denotes the usual orthonormal
basis of $\mathbb{R}^{n}$. Let us denote
$\alpha_{ij}^{\pm}:=\alpha_{i}\pm\alpha_{j}$ for $1\leq i<j\leq n$.
Furthermore, the negative roots are denoted by
$\hat{\alpha}_{ij}^{\pm}=-\left(\alpha_{i}\pm\alpha_{j}\right)$ and
$\hat{\alpha}_{k}$. In particular, $\hat{\alpha}_{ij}^{-}=\alpha_{ji}^{-}.$
Notice that the simple roots of $B_{n}$ are given by
$\Delta_{B_{n}}=\\{\alpha_{ss+1}^{-},\alpha_{n}:\text{ }1\leq s\leq n-1\\}.$

The roots corresponding to those root vectors are given by

(33) $\displaystyle
R\left(\epsilon_{ij}^{-}\right)=\alpha_{ij}^{-}:=\sum_{s=i}^{j-1}\alpha_{ss+1}^{-},\quad
R\left(\epsilon_{ij}^{+}\right)=\alpha_{ij}^{+}:=\sum_{s=i}^{j-1}\alpha_{ss+1}^{-}+2\sum_{s=j}^{n-1}\alpha_{ss+1}^{-}+2\alpha_{n},\quad
R\left(\epsilon_{j}\right):=\sum_{s=j}^{n-1}\alpha_{ss+1}^{-}+\alpha_{n}.$

Note that $\epsilon_{12}^{+}$ is the maximal length of the root in the system.
Moreover, $\alpha_{in}^{+}=\alpha_{in}^{-}+2\alpha_{n}$ for all $1\leq i<n$.
Since all roots are $ad(h_{i})$-daigonalizable for all
$h_{i}\in\mathfrak{h}^{*},$ from $\eqref{eq:roots},$ we will deduce the
following PDEs

(34)
$\displaystyle\\{h_{i},\epsilon_{jk}^{-}\\}_{o}=\mu_{j,k}^{i}\epsilon_{jk}^{-}\text{
and
}\\{h_{i},\epsilon_{j}\\}_{o}=\left(\mu_{j,n}^{i}+\mu_{n}^{i}\right)\epsilon_{j}=\mu_{j,j}^{i}\epsilon_{j},$

where

(35) $\displaystyle\mu_{j,j+1}^{j\pm
1}=-1,\quad\mu_{j,j+1}^{j}=2,\quad\mu_{n}^{n-1}=-2=-\mu_{n}^{n},\quad\mu_{j,j+1}^{l}=0$

for any $l\neq j,j\pm 1$ and $1\leq j\leq n-1.$

We now proceed to classify the generators in $\mathcal{Q}_{B_{n}}(d)$ for all
$n\geq 1.$ From [6], we know that the indecomposable monomials consisting of
the root vectors in $\epsilon_{ii+1}^{-}$ can be realized as $k$-cycles in
$S_{n}$ which is the Weyl group of $A_{n}.$ Since $\Phi_{B_{n}}$ contains the
long and short roots so that the Cartan matrix is no longer a permutation
matrix, it is reasonable to consider the Weyl group of $B_{n}$. That is,
$\mathcal{W}_{B_{n}}:=S_{n}\rtimes\left(\mathbb{Z}_{2}\right)^{n}.$ This will
lead to a distinct way, comparing the Cartan centralizer in $A_{n},$ to
classify indecomposable monomials in $\mathcal{Q}_{B_{n}}(d)$. Consider the
symmetric algebra of $B_{n}$ for any $n\geq 1$, by the PBW theorem and
arguments in Section 3, $\mathcal{Q}_{B_{n}}(d)$ is spanned by the Cartan
subalgebra and the set

(36)
$\displaystyle\left\\{\left(\epsilon_{12}^{-}\right)^{l_{1}}\cdots\left(\epsilon_{12}^{+}\right)^{l_{\frac{n(n-1)}{2}+1}}\cdots\left(\epsilon_{1}\right)^{l_{n(n-1)+1}}\cdots\left(\epsilon_{n}\right)^{l_{n^{2}}}\left(\epsilon_{21}^{-}\right)^{m_{1}}\cdots\left(\hat{\epsilon}_{n}\right)^{m_{n^{2}}}:\sum_{i}(l_{i}-m_{i})\gamma_{i}=0\right\\}.$

Notice that span of $\eqref{eq:ge}$ will give us a symmetric algebra
$S(\mathfrak{u})$. Taking into account the Weyl group $\mathcal{W}_{B_{n}},$
we have the following definition.

######  Definition 4.1.

Let $\mathcal{Q}_{B_{n}}(d)$ be a polynomial algebra in Cartan centralizer.
Let $\textbf{x}_{1}=\left(\epsilon_{12}^{-},\ldots,\epsilon_{n-1n}^{-}\right)$
and
$\textbf{x}_{2}=\left(\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{+},\epsilon_{1},\ldots,\hat{\epsilon}_{n}\right)$.
Suppose that

$\displaystyle S_{n}\text{-span}:=$ $\displaystyle\text{
}\mathrm{span}\left\\{\left(\epsilon_{12}^{-}\right)^{l_{1}}\cdots\left(\epsilon_{n-1n}^{-}\right)^{l_{n(n-1)}}:\textbf{l}=\left(l_{1},\ldots,l_{n(n-1)}\right)\in\mathbb{N}_{0}^{n(n-1)}\right\\},$
(37) $\displaystyle\mathbb{Z}_{2}\text{-span}:=$ $\displaystyle\text{
}\mathrm{span}\left\\{\left(\epsilon_{12}^{+}\right)^{m_{1}}\cdots\left(\hat{\epsilon}_{n-1n}^{+}\right)^{m_{2n(n-1)}}\prod_{a,b}\left(\epsilon_{a}\right)^{\ell_{a}}\left(\hat{\epsilon}_{b}\right)^{\ell_{b}}:\text{
}m_{w},m_{u},\ell_{a},\ell_{b}\in\mathbb{N}_{0}\right\\}.$

The polynomials in $\mathcal{Q}_{B_{n}}(d)$ have the form of
$p(\textbf{x}_{B})=f(\textbf{h})g(\textbf{x}_{1})m(\textbf{x}_{2}),$ where
$\textbf{x}_{B}=\left(h_{1},\ldots,h_{n},\epsilon_{12}^{-},\ldots,\epsilon_{n-1n}^{-},\right.$
$\left.\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{-},\epsilon_{1},\ldots,\hat{\epsilon}_{n}\right)$
is a coordinate in $\mathfrak{so}^{*}(2n+1,\mathbb{C})$, $g(\textbf{x}_{1})\in
S_{n}$-span and $m(\textbf{x}_{2})\in\mathbb{Z}_{2}$-span. We call the root
vectors in $g(\textbf{x}_{1})$ as permutation type of roots and
$m(\textbf{x}_{2})$ component as $\mathbb{Z}_{2}$-type of roots.

######  Remark 4.2.

(i) Notice that we can split $\mathbb{Z}_{2}$-span into two parts. By the PBW-
theorem, we could write $\mathbb{Z}_{2}$-span
$:=\mathbb{Z}_{2}^{L}\text{-span}\otimes\mathbb{Z}_{2}^{S}\text{-span}$, where

$\displaystyle\mathbb{Z}_{2}^{S}\text{-}\mathrm{span}=$ $\displaystyle\text{
}\mathrm{span}\left\\{\prod_{a,b}\left(\epsilon_{a}\right)^{\ell_{a}}\left(\hat{\epsilon}_{b}\right)^{\ell_{b}}:\text{
}\ell_{a},\ell_{b}\in\mathbb{N}_{0}\right\\};$
$\displaystyle\mathbb{Z}_{2}^{L}\text{-}\mathrm{span}=$ $\displaystyle\text{
}\mathrm{span}\left\\{\left(\epsilon_{12}^{+}\right)^{m_{1}}\cdots\left(\epsilon_{n-1n}^{+}\right)^{m_{n(n-1)}}\left(\hat{\epsilon}_{12}^{+}\right)^{m_{n(n-1)+1}}\cdots\left(\hat{\epsilon}_{n-1n}^{+}\right)^{m_{2n(n-1)}}:\text{
}m_{w},m_{u}\in\mathbb{N}_{0}\right\\}.$

Hence, we can further decompose the root vectors of $\textbf{x}_{2}$ into
$\textbf{x}_{L}=\left(\epsilon_{12}^{+},\ldots,\epsilon_{n-1n}^{+},\hat{\epsilon}_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{+}\right)$
and
$\textbf{x}_{S}=\left(\epsilon_{1},\ldots,\epsilon_{n},\hat{\epsilon}_{1},\ldots,\hat{\epsilon}_{n}\right)$
such that $m(\textbf{x}_{2})=m_{1}(\textbf{x}_{L})m_{2}(\textbf{x}_{S})$,
where $m_{1}(\textbf{x}_{L})\in\mathbb{Z}_{2}^{L}$-span and
$m_{2}(\textbf{x}_{S})\in\mathbb{Z}_{2}^{S}$-span.

(ii) Suppose that
$p\left(\epsilon_{ij}^{-},\textbf{x}_{2}\right)=\epsilon_{ij}^{-}m(\textbf{x}_{2})\in\mathcal{Q}_{B_{n}}(d)$
with $m(\textbf{x}_{2})\in\mathbb{Z}_{2}$-span and $1\leq i\neq j\leq n$. It
turns out that there exists a partition $\left(\lambda_{j}\right)_{j=1}^{l}$
such that
$\epsilon_{i\lambda_{1}}^{-}\epsilon_{\lambda_{1}\lambda_{2}}^{-}\cdots\epsilon_{\lambda_{l}j}^{-}m(\textbf{x}_{2})\in\mathcal{Q}_{B_{n}}(d)$
is indecomposable. In what follows, we denote the equivalence classes of the
permutation type of roots such that $p$ is indecomposable by

(38)
$\displaystyle\left[\epsilon_{ij}^{-}\right]:=\left\\{\epsilon_{i\lambda_{1}}^{-}\epsilon_{\lambda_{1}\lambda_{2}}^{-}\cdots\epsilon_{\lambda_{l}j}^{-}:1\leq\lambda_{j}\leq
n,\text{ }\lambda_{j}\in\mathbb{N}\right\\}.$

In other words, if
$\epsilon_{ij}^{-}m(\textbf{x}_{2})\in\mathcal{Q}_{B_{n}}(d)$ is
indecomposable, then
$\left[\epsilon_{ij}^{-}\right]m(\textbf{x}_{2})\in\mathcal{Q}_{B_{n}}(d)$ is
also indecomposable.

(iii) By Definition 4.1, under a coordinate
$\textbf{x}_{B}=\left(h_{1},\ldots,h_{n},\epsilon_{12}^{-},\ldots,\epsilon_{n-1n}^{-},\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{-},\epsilon_{1},\right.$
$\left.\ldots,\hat{\epsilon}_{n}\right)$ of $B_{n}$, an element in
$\mathcal{Q}_{B_{n}}(d)$ has the generic form of
$p(\textbf{x}_{B})=f(\textbf{h})g(\textbf{x}_{1})m_{1}(\textbf{x}_{L})m_{2}(\textbf{x}_{S}).$
However, from the algebraic structures of $\mathcal{Q}_{B_{n}}(d)$,
$S(\mathfrak{h})$ is the first layer in $\mathcal{Q}_{B_{n}}(d)$. Hence, if
$p(\textbf{x}_{B})$ contains $f(\textbf{h})$ in its expression, it must be
decomposable. Therefore, throughout this paper, we will study the form of
$p(\textbf{x}_{B}^{\prime})=g(\textbf{x}_{1})m_{1}(\textbf{x}_{L})m_{2}(\textbf{x}_{S})$,
where
$\textbf{x}_{B}^{\prime}=\left(\epsilon_{12}^{-},\ldots,\epsilon_{n-1n}^{-},\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{-},\epsilon_{1},\ldots,\hat{\epsilon}_{n}\right).$
In what follows, to avoid abuse of notations, we will take
$\textbf{x}_{B}^{\prime}:=\textbf{x}_{B}$ such that
$p(\textbf{x}_{B}^{\prime}):=p(\textbf{x}_{B}).$

In this paragraph, we find the explicit expression of the generators in
$U(B_{n})^{\mathfrak{h}}$. Define the partition of indices by

(39) $\displaystyle\mathcal{S}^{+}=\left\\{(i_{s},j_{s}):i_{s}<j_{s},\text{
}1\leq i_{s},j_{s}\leq n\right\\}\text{ and
}\mathcal{S}^{-}=\left\\{(i_{s},j_{s}):i_{s}>j_{s},\text{ }1\leq
i_{s},j_{s}\leq n\right\\}.$

Those ordered sets in $\eqref{eq:indices}$ will help us to distinguish the
positive or negative roots that appear in $S_{n}$-span. Using the properties
of the root system, it is easy to observe the following facts:

(i) For all $1\leq i,j\leq n,$ the monomials
$\epsilon_{ij}^{-}\epsilon_{ji}^{-},$
$\epsilon_{ij}^{+}\hat{\epsilon}_{ij}^{+}$ and
$\epsilon_{i}\hat{\epsilon}_{i}$ are indecomposable generators that span
$\mathcal{Q}_{2}\subset\mathcal{Q}_{B_{n}}(d)$.

(ii) For some $r,q,t\in\mathbb{N}_{0}$ and all pairs in $\mathcal{S}^{\pm}$,
the monomials $\prod_{s}^{r}\epsilon_{i_{s}j_{s}}^{-}$,
$\prod_{a}^{q}\epsilon_{s_{a}}$, $\prod_{w}^{t}\epsilon_{k_{w}l_{w}}^{+}$,
$\prod_{a}^{q}\epsilon_{s_{a}}\left(\hat{\epsilon}_{n}\right)^{q}$ and
$\prod_{w}^{t}\epsilon_{k_{w}l_{w}}^{+}\left(\hat{\epsilon}_{n}\right)^{2t}$are
in $S(\mathfrak{g}^{+}).$ Hence they are not in $\mathcal{Q}_{B_{n}}(d)$.

(iii) Suppose that
$p(\textbf{x}_{B})=g(\textbf{x}_{1})m(\textbf{x}_{2})\in\mathcal{Q}_{B_{n}}(d)$
with $g(\textbf{x}_{1})\in S_{n}$-span and
$m(\textbf{x}_{2})\in\mathbb{Z}_{2}$-span. We deduce that $\alpha_{n}\notin
R\left(p\right)$ and
$\hat{p}:=p(\hat{\textbf{x}}_{B})\in\mathcal{Q}_{B_{n}}(d)$. Hence $\deg
m(\textbf{x}_{2})\in 2\mathbb{N}.$ Here $\hat{\textbf{x}}_{B}$ defines a
negative root vector.

By definition 4.1, we can assume that

(40) $\displaystyle
p\left(\epsilon_{i_{1}j_{1}}^{-},\ldots,\hat{\epsilon}_{k_{t}l_{t}}^{+},\epsilon_{e_{1}},\ldots,\hat{\epsilon}_{e_{q}}\right)=\prod_{c=1}^{r}\epsilon_{i_{c}j_{c}}^{-}\prod_{w=1}^{\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{t}\hat{\epsilon}_{k_{u}l_{u}}^{+}\prod_{a=1}^{\nu}\epsilon_{s_{a}}\prod_{b=\nu+1}^{q}\hat{\epsilon}_{s_{b}}\in\mathcal{Q}_{B_{n}}(d)$

In particular, we define

$\displaystyle
g\left(\epsilon_{i_{1}j_{1}}^{-},\ldots,\epsilon_{i_{r}j_{r}}^{-}\right)=$
$\displaystyle\text{ }\prod_{c=1}^{r}\epsilon_{i_{c}j_{c}}^{-}\in
S_{n}\text{-span};$ $\displaystyle
m_{1}\left(\epsilon_{k_{1}l_{1}}^{+},\ldots,\epsilon_{k_{\mu}l_{\mu}}^{+},\hat{\epsilon}_{k_{\mu+1}l_{\mu+1}},\ldots,\hat{\epsilon}_{k_{t}l_{t}}^{+}\right)=$
$\displaystyle\text{
}\prod_{w=1}^{\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{t}\hat{\epsilon}_{k_{u}l_{u}}^{+}\in\mathbb{Z}_{2}^{L}\text{-span};$
$\displaystyle
m_{2}\left(\epsilon_{s_{1}},\ldots,\epsilon_{s_{\nu}},\hat{\epsilon}_{\nu+1},\ldots,\hat{\epsilon}_{q}\right)=$
$\displaystyle\text{
}\prod_{a=1}^{\nu}\epsilon_{s_{a}}\prod_{b=\nu+1}^{q}\hat{\epsilon}_{s_{b}}\in\mathbb{Z}_{2}^{S}\text{-span}.$

Here $r,t,q,\mu,\nu\in\mathbb{N}_{0}$ are undetermined constants. Roughly
speaking, these constants are constrained by the rank of Lie algebras. By the
definition above and the basis of the symmetric algebra $S(B_{n}),$ we can
provide the upper bound for those constants. That is, $r\leq n(n-1)$,
$\mu\leq\frac{n(n-1)}{2}$ and $\nu\leq n.$ To accurate those upper bound, we
will present some examples in Section 5 to perform the precise relation
between $\mathrm{rank}\,\mathfrak{g}$ and these constants. Now, back to the
progress on finding the explicit form of $p.$ Since $\alpha_{n}\notin
R\left(p\right),$ the corresponding roots to $p\in\mathcal{Q}_{B_{n}}(d)$ have
the form of

(41) $\displaystyle
R\left(p\right)=\sum_{c}^{r}\alpha_{i_{c}j_{c}}^{-}+\sum_{w=1}^{\mu}\left(\alpha_{k_{w}n}^{-}+\alpha_{l_{w}n}^{-}\right)+\sum_{w=\mu+1}^{t}\left(\alpha_{nk_{w}}^{-}+\alpha_{nl_{w}}^{-}\right)+\sum_{a=1}^{\nu}\alpha_{s_{a}n}^{-}+\sum_{b=\nu+1}^{q}\alpha_{ns_{b}}^{-}=0.$

Hence, to make the identity $\eqref{eq:r41ida}$ vanishing, it is reasonable to
consider all the possible linear combinations of $\alpha_{k_{w},l_{w}}^{+}$
and $\alpha_{s_{a}}$ such that the root vectors $\alpha_{n}$ vanish. In other
words, we need to find

(42) $\displaystyle\alpha_{n}\notin
R(m)=\sum_{w=1}^{t}\gamma(m_{w})\alpha_{k_{w},l_{w}}^{+}+\sum_{a=1}^{q}\gamma(\ell_{a})\alpha_{s_{a}}$

such that
$\sum_{s=1}^{r}\alpha_{i_{s},j_{s}}^{-}=-\left(\sum_{w=1}^{t}\gamma(m_{w})\alpha_{k_{w},l_{w}}^{+}+\sum_{a=1}^{q}\gamma(\ell_{a})\alpha_{s_{a}}\right),$
where $\gamma(m_{w}),\gamma(\ell_{a})$ are some constant functions. Using
Definition 4.1 and the following remark, for any
$p\in\mathcal{Q}_{\mathfrak{g}}(d),$ it is natural to consider $5$ different
cases:

(a) $p$ equals $g$, $m_{1}$ or $m_{2};$ (b) $p=gm_{1};$ (c) $p=gm_{2}$; (d)
$p=m_{1}m_{2}$; (e) $p=gm_{1}m_{2}.$

In order to proceed with the classification, we first need to ensure that
$R(p)=0$ holds for any $p$ in the five cases above. In other words,
$\alpha_{n}\notin R(m)$. This constraint provides the generic forms of $m_{1}$
and $m_{2}$ respectively. See, $\eqref{eq:classif}$ and $\eqref{eq:classif2}$
in Section $\ref{5}.$ Then we only need to analyze all the possible
permutation roots that cover all the roots in
$R(m_{1})=\sum_{w=1}^{\mu}\left(\alpha_{k_{w}n}^{-}+\alpha_{l_{w}n}^{-}\right)+\sum_{w=\mu+1}^{t}\left(\alpha_{nk_{w}}^{-}+\alpha_{nl_{w}}^{-}\right)+(2\mu-2t)\alpha_{n}$
and
$R(m_{2})=\sum_{a=1}^{\nu}\alpha_{s_{a}n}^{-}+\sum_{b=\nu+1}^{q}\alpha_{ns_{b}}^{-}+(\nu-q)\alpha_{n}$
respectively. Note that the formal proof of the classification is not trivial
and will be provided in [5]. Here, we only present the final results as
follows:

(a) Suppose that $p(\textbf{x}_{B})$ equals $g(\textbf{x}_{1}),$
$m_{1}(\textbf{x}_{L})$ or $m_{2}(\textbf{x}_{S})$ with $\deg m_{i}>2$ for all
$i=1,2$. Then
$\mathcal{Q}_{A_{n}}(n):=\mathrm{span}\left\\{h_{k},\left[\epsilon_{ij}^{-}\right]\epsilon_{ji}^{-}:1\leq
i,j,k\leq n\right\\}\subset\mathcal{Q}_{B_{n}}(d),$
$m_{1}\left(\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{l_{t}k_{1}}^{+}\right)=\epsilon_{k_{1}l_{1}}^{+}\hat{\epsilon}_{l_{1}l_{2}}^{+}\cdots\epsilon_{l_{t-1}l_{t}}^{+}\hat{\epsilon}_{l_{t}k_{1}}^{+},$
$\hat{m}_{1}\in\mathcal{Q}_{2h}$ and
$m_{2}\left(\epsilon_{s_{1}},\ldots,\epsilon_{s_{\nu}},\hat{\epsilon}_{s_{\nu+1}},\ldots,\hat{\epsilon}_{s_{2\nu}}\right)$
are decomposable, where $1\leq k_{1},l_{1},\ldots,l_{t}\leq n.$

Let
$\mathcal{SK}_{1}:=\left(S_{n}\text{-span}\otimes\mathbb{Z}_{2}^{L}\text{-span}\right)\cap\mathcal{Q}_{B_{n}}(d)$.

(b1) Suppose that

(43) $\displaystyle
p\left(\epsilon_{i_{1}j_{1}}^{-},\ldots,\epsilon_{i_{r}j_{r}}^{-},\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{t}l_{t}}^{+}\right)=\prod_{s=1}^{r}\epsilon_{i_{s}j_{s}}^{-}\prod_{w=1}^{\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{t}\hat{\epsilon}_{k_{u}l_{u}}^{+}\in\mathcal{SK}_{1}$

with $1\leq k_{w},l_{w}\neq k_{u},l_{u}\leq n$. Then $t=2\mu$, moreover, there
exists

(44)
$\displaystyle\left[\epsilon_{k_{u_{1}}k_{1}}^{-}\right]\left[\epsilon_{l_{u_{2}}l_{1}}^{-}\right]\cdots\left[\epsilon_{k_{u_{\mu}}k_{\mu}}^{-}\right]\left[\epsilon_{l_{u_{1}}l_{\mu}}^{-}\right]\in
S_{n}\text{-span}$

with $n\geq 4$ such that $\eqref{eq:id630}$ are indecomposable, where
$u_{i}\neq u_{j}\in U_{\mu}:=\\{\mu+1,\ldots,2\mu\\}$ is a reordering of
indices.

(b2) Suppose that

$\displaystyle
m_{1}^{x}\left(\epsilon_{k_{1}l_{1}}^{+},\ldots,\epsilon_{k_{\mu}l_{\mu}}^{+},\hat{\epsilon}_{k_{\mu+1}l_{\mu+1}}^{+},\ldots,\epsilon_{l_{2\mu-1}l_{2\mu}}^{+}\right)$
$\displaystyle=\prod_{w=1}^{\xi}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{\mu+\xi}\hat{\epsilon}_{k_{u}l_{u}}^{+}\prod_{a=\xi+1}^{\mu}\epsilon_{k_{a}l_{a}}^{+}\prod_{b=\mu+\xi+1}^{2\mu}\hat{\epsilon}_{l_{a}l_{b}}^{+}$
$\displaystyle
m_{1}^{y}\left(\epsilon_{k_{1}l_{1}}^{+},\ldots,\epsilon_{k_{\mu}l_{\mu}}^{+},\epsilon_{l_{1}l_{\mu+1}}^{+},\ldots,\hat{\epsilon}_{l_{\mu}l_{2\mu}}\right)$
$\displaystyle=\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{l_{w}l_{u}}^{+}$

$k_{w},l_{u}\notin\left\\{l_{w}:1\leq w\leq\mu\right\\}$. Then there exists
some $\left[g_{1}\right]\in S_{n}$-span with $n\geq 4$ and
$\left[g_{2}\right]\in S_{n}$-span with $n\geq 3$ induced from
$\eqref{eq:id631}$ such that $[g_{1}]m_{1}^{x}$ and $[g_{2}]m_{1}^{y}$ are
indecomposable.

(c) Let
$\mathcal{SK}_{2}:=\left(S_{n}\text{-span}\otimes\mathbb{Z}_{2}^{S}\text{-span}\right)\cap\mathcal{Q}_{B_{n}}(d)$.
Suppose that

(45) $\displaystyle
p\left(\epsilon_{i_{1}j_{1}}^{-},\ldots,\epsilon_{i_{r}j_{r}}^{-},\epsilon_{s_{1}},\ldots,\hat{\epsilon}_{s_{q}}\right)=\prod_{s=1}^{r}\epsilon_{i_{s}j_{s}}^{-}\prod_{a}^{\nu}\epsilon_{s_{a}}\prod_{b=\nu+1}^{q}\hat{\epsilon}_{s_{b}}\in\mathcal{SK}_{2}.$

Then there exists $\left[\prod_{s=1}^{r}\epsilon_{i_{s}j_{s}}^{-}\right]\in
S_{n}$-span such that $p$ defined in $\eqref{eq:sk2}$ are indecomposable if
and only if $\deg m_{2}=2.$ Moreover, $\mathcal{SK}_{2}$ is spanned by

(46)
$\displaystyle\left[\epsilon_{ij}^{-}\right]\hat{\epsilon}_{i}\epsilon_{j},\text{
and }\left[\epsilon_{ji}^{-}\right]\epsilon_{i}\hat{\epsilon}_{j},$

where $1\leq i,j\leq n.$

(d) Let
$\mathcal{K}_{1}\mathcal{K}_{2}:=\left(\mathbb{Z}_{2}^{S}\text{-span}\otimes\mathbb{Z}_{2}^{L}\text{-span}\right)\cap\mathcal{Q}_{B_{n}}(d)$.
Then

$\displaystyle
m\big{(}\textbf{x}_{L_{1}},\hat{\epsilon}_{b_{1}},\hat{\epsilon}_{b_{2}}\big{)}$
$\displaystyle=\epsilon_{kl}^{+}\prod_{1\leq
w\leq\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{\mu+1\leq u\leq
2\mu}\hat{\epsilon}_{k_{u}l_{u}}^{+}\hat{\epsilon}_{b_{1}}\hat{\epsilon}_{b_{2}},$
$\displaystyle m(\textbf{x}_{L_{2}},\epsilon_{a},\hat{\epsilon}_{b})$
$\displaystyle=\prod_{1\leq w\leq\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{\mu+1\leq
u\leq 2\mu}\hat{\epsilon}_{k_{u}l_{u}}^{+}\epsilon_{a}\hat{\epsilon}_{b}$

are indecomposable, where $a\neq
b,b_{1},b_{2}\in\left\\{k_{w},l_{w},l_{u},l_{u}:1\leq w\leq\mu,\text{
}\mu+1\leq u\leq 2\mu,\text{ }1\leq k_{w},l_{w},l_{u},l_{u}\leq n\right\\}$,
$\textbf{x}_{L_{1}}=\left(\epsilon_{kl}^{+},\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{2\mu}l_{2\mu}}^{+}\right)$
and
$\textbf{x}_{L_{2}}=\left(\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{2\mu}l_{2\mu}}^{+}\right)$.

(e) Suppose that there exists a $\left[g(\textbf{x}_{1})\right]\in S_{n}$-span
with $n\geq 3$ such that
$\left[g(\textbf{x}_{1})\right]m_{1}(\textbf{x}_{L})m_{2}(\textbf{x}_{S})\in\mathcal{Q}_{B_{n}}(d)$
is indecomposable. Then $\deg m_{2}(\textbf{x}_{S})=2.$ Conversely, if $\deg
m_{2}(\textbf{x}_{S})=2,$ then there exist some
$\left[g(\textbf{x}_{I})\right],\left[g(\textbf{x}_{II})\right]\in S_{n}$-span
such that

$\displaystyle
p_{I}\left(\textbf{x}_{I},\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{2\mu}l_{2\mu}}^{+},\epsilon_{a},\hat{\epsilon}_{b}\right)=$
$\displaystyle\left[g(\textbf{x}_{I})\right]\prod_{w=1}^{\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{2\mu}\hat{\epsilon}_{k_{u}l_{u}}^{+}\epsilon_{a}\hat{\epsilon}_{b};$
$\displaystyle
p_{II}\left(\textbf{x}_{II},\epsilon_{kl}^{+},\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{2\mu}l_{2\mu}}^{+},\hat{\epsilon}_{b_{1}},\hat{\epsilon}_{b_{2}}\right)=$
$\displaystyle\left[g(\textbf{x}_{II})\right]\epsilon_{kl}^{+}\prod_{w=1}^{\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{2\mu}\hat{\epsilon}_{k_{u}l_{u}}^{+}\hat{\epsilon}_{b_{1}}\hat{\epsilon}_{b_{2}}$

are indecomposable monomials. Here $\textbf{x}_{I}$ and $\textbf{x}_{II}$ are
distinct permutation root vectors, $1\leq a\neq b\leq n$ and $1\leq
b_{1},b_{2}\leq n.$

In this way, we can define a polynomial algebra
$\mathcal{Q}_{B_{n}}(2n-1)=\mathrm{span}\text{ }\textbf{Q}_{2n}$ under a
bilinear map $\\{\cdot,\cdot\\}$. For the rest of the section, we will provide
a brief description of the Cartan commutant for the cases $C_{n}$ and $D_{n}$.

###  4.2. The explicit generator of $U(D_{n})^{\mathfrak{h}}$

A Cartan subalgebra of $D_{n}$ is defined by
$\mathfrak{h}=\left\\{\mathrm{diag}(a_{1},\ldots,a_{n},-a_{1},\right.$
$\left.\ldots,-a_{n}):a_{i}\in\mathbb{R},\text{ }1\leq i\leq n\right\\}.$ For
any $H\in\mathfrak{h},$ we can write
$H=\sum_{j=1}^{n}a_{j}\left(E_{j,j}-E_{n+j,n+j}\right).$ A direct computation
shows that

(47) $\displaystyle\begin{matrix}[H_{i},E_{i,i+1}]_{o}=2E_{i,i+1},\text{
}[H_{i-1},E_{i,i+1}]_{o}=-E_{i,i+1},\text{
}[H_{i+1},E_{i,i+1}]_{o}=-E_{i,i+1},\\\
[H_{n-2},E_{n-2,n-1}]_{o}=-E_{n-2,n-1},\text{
}[H_{n-2},E_{n-1,n}]_{o}=-E_{n-1,n},\text{
}[H_{n},E_{n-1,n}]_{o}=0=[H_{j},E_{i,i+1}]_{o},\end{matrix}$

where $j\neq i\pm 1$ and $i=1,\ldots,n-2.$ It is clear that the eigenvalues
from $\eqref{eq:eigen22}$ form the column vectors of the Cartan matrix
$C_{D},$ which is given by

$\displaystyle C_{D}=\begin{pmatrix}2&-1&\ldots&0&0&0\\\ -1&2&\ldots&0&0&0\\\
\vdots&\ldots&\vdots&\vdots&\vdots&\vdots\\\ 0&0&\ldots&2&-1&-1\\\
0&0&\ldots&-1&2&0\\\ 0&0&\ldots&-1&0&2\end{pmatrix}.$

The dual space $\mathfrak{so}^{*}(2n,\mathbb{C})$ admits a coordinate
$\textbf{x}_{D}=\left(h_{1},\ldots,h_{n},\epsilon_{12}^{-},\ldots,\epsilon_{n-1n}^{-},\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{-}\right),$
where $\hat{\cdot}$ defines a negative root. Notice that
$\Delta_{\mathfrak{so}(2n,\mathbb{C})}=\left\\{\alpha_{ss+1}^{-},\alpha_{n}+\alpha_{n-1}\right\\}_{s=1}^{n-1}$
forms a basis of the root system of $\mathfrak{so}(2n,\mathbb{C}).$ The
corresponding roots to root vectors are

(48) $\displaystyle
R\left(\epsilon_{ij}^{-}\right)=\sum_{s=i}^{j-1}\alpha_{ss+1}^{-}\text{ and
}R\left(\epsilon_{ij}^{+}\right)=\alpha_{in-1}^{-}+\alpha_{jn-1}^{-}+\alpha_{n-1n}^{-}+\alpha_{n-1}+\alpha_{n}.$

Observe that $\alpha_{n-1n}^{+}=\alpha_{n-1}+\alpha_{n}$ and
$\alpha_{in-1}^{+}=\alpha_{in}^{-}+\alpha_{n-1}+\alpha_{n}$. Considering the
symmetric space of $D_{n}$, we need to find the set of generators

(49)
$\displaystyle\left\\{\left(\epsilon_{12}^{-}\right)^{l_{1}}\cdots\left(\epsilon_{12}^{+}\right)^{l_{\frac{n(n-1)}{2}+1}}\cdots\left(\epsilon_{n-1n}^{+}\right)^{l_{n(n-1)+1}}\left(\epsilon_{21}^{-}\right)^{m_{1}}\cdots\left(\hat{\epsilon}_{n-1n}^{+}\right)^{m_{n^{2}}}:\sum_{i}(l_{i}-m_{i})\alpha_{i}=0\right\\}.$

Hence, similar to the permutation and long roots in
$\Phi_{\mathfrak{so}(2n+1,\mathbb{C})},$ we can separate the monomials into
the tensor product of distinct root vectors.

######  Definition 4.3.

Let $\mathfrak{g}=\mathfrak{so}(2n,\mathbb{C})$ and $\mathfrak{g}^{*}$ be its
dual space with the coordinate
$\textbf{x}_{D}=\left(h_{1},\ldots,h_{n},\epsilon_{12}^{-},\right.$
$\left.\ldots,\epsilon_{nn-1}^{-},\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{+}\right).$
Let $\textbf{x}_{1}=\left(\epsilon_{12}^{-},\ldots,\epsilon_{nn-1}^{-}\right)$
and
$\textbf{x}_{L}=\left(\epsilon_{12}^{+},\ldots,\hat{\epsilon}_{n-1n}^{+}\right).$
For any
$p(\textbf{x}_{D})=g(\textbf{x}_{1})m_{1}(\textbf{x}_{L})\in\mathcal{Q}_{D_{n}}(d),$
suppose that $g(\textbf{x}_{1})\in S_{n}$-span and

$\mathbb{Z}_{2}^{L}\text{-}\mathrm{span}=\text{
}\mathrm{span}\left\\{\left(\epsilon_{12}^{+}\right)^{m_{1}}\cdots\left(\epsilon_{n-1n}^{+}\right)^{m_{n(n-1)}}\left(\hat{\epsilon}_{12}^{+}\right)^{m_{n(n-1)+1}}\cdots\left(\hat{\epsilon}_{n-1n}^{+}\right)^{m_{2n(n-1)}}:\text{
}m_{w},m_{u}\in\mathbb{N}_{0}\right\\}.$

We call the subspaces spanned by $m_{1}(\textbf{x}_{L})$ as
$\mathbb{Z}_{2}^{L}$-span.

By definition, the monomial in $\mathcal{Q}_{D_{n}}(d^{\prime})$ has the form
of

(50) $\displaystyle
p\left(\epsilon_{i_{1}j_{1}}^{-},\ldots,\epsilon_{i_{r}j_{r}}^{-},\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{t}l_{t}}^{+}\right)=g\left(\epsilon_{i_{1}j_{1}}^{-},\ldots,\epsilon_{i_{r}j_{r}}^{-}\right)m_{1}\left(\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{t}l_{t}}^{+}\right),$

where
$g\left(\epsilon_{i_{1}j_{1}}^{-},\ldots,\epsilon_{i_{r}j_{r}}^{-}\right)=\prod_{s=1}^{r}\epsilon_{i_{s}j_{s}}^{-}\in
S_{n}$-span and
$m_{1}\left(\epsilon_{k_{1}l_{1}}^{+}\ldots\hat{\epsilon}_{k_{t}l_{t}}^{+}\right)=\prod_{w=1}^{\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{t}\hat{\epsilon}_{k_{u}l_{u}}^{+}\in\mathbb{Z}_{2}^{L}$-span.
Similar to the case $B_{n}$, we need to find all the possible combination of
the terms $\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}$ such that
the constant term $\alpha_{n-1n}^{-}+\alpha_{n-1}+\alpha_{n}$ vanishing from
$R\left(m_{1}\right).$ By definition,

$R(g)=\sum_{w=1}^{\mu}\left(\alpha_{k_{w}n-1}^{-}+\alpha_{l_{w}n-1}^{-}\right)+\sum_{u=\mu+1}^{2\mu}\left(\alpha_{n-1k_{u}}^{-}+\alpha_{n-1l_{u}}^{-}\right).$

Similar to Subsection $\ref{4.1},$ it can be shown that that the
classification of different types of the monomials are given by:

(a) Suppose that a monomial $p\in\mathcal{Q}_{D_{n}}(d^{\prime})\cap
S_{n}$-span or $\mathcal{Q}_{D_{n}}(d^{\prime})\cap\mathbb{Z}_{2}^{L}$-span.
Then $\mathcal{Q}_{A_{n}}(n)\subset\mathcal{Q}_{D_{n}}(d^{\prime})$ and
$m_{1}\left(\epsilon_{k_{1}l_{2}}^{+},\ldots,\hat{\epsilon}_{l_{t}k_{1}}^{+}\right)=\epsilon_{k_{1}l_{2}}^{+}\hat{\epsilon}_{l_{2}l_{3}}^{+}\cdots\epsilon_{l_{t-1}l_{t}}^{+}\hat{\epsilon}_{l_{t}k_{1}}^{+}\in\mathcal{Q}_{2h}$
and $\hat{m}_{1}$ are indecomposable monomials in
$\mathcal{Q}_{D_{n}}(d^{\prime})$, where $1\leq k_{1},l_{1},\ldots,l_{t}\leq
n.$

(b) Let
$\mathcal{SK}_{1}:=\left(S_{n}\text{-span}\otimes\mathbb{Z}_{2}^{L}\text{-span}\right)\cap\mathcal{Q}_{D_{n}}(d^{\prime})$.
Then there exists some $\prod_{s=1}^{r}\epsilon_{i_{s}j_{s}}^{-}\in
S_{n}$-span with $n\geq 3$ such that

(51)
$\displaystyle\prod_{s=1}^{r}\epsilon_{i_{s}j_{s}}^{-}\prod_{w=1}^{\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{u=\mu+1}^{2\mu}\hat{\epsilon}_{k_{u}l_{u}}^{+}\in\mathcal{SK}_{1}$

are indecomposable.

Therefore, it is sufficient to claim that
$\mathcal{Q}_{D_{n}}(2n-4)=\mathbb{C}\langle\textbf{Q}_{2n-2}\rangle$ form a
polynomial algebra under the bilinear bracket $\\{\cdot,\cdot\\}$, and an
inclusion relation
$\mathcal{Q}_{D_{n}}(2n-4)\subset\mathcal{Q}_{B_{n}}(2n-1)$. Notice that if
$n=2,$ we will have that $\mathcal{Q}_{D_{2}}(0)$ is Abelian, as shown in the
example below.

######  Example 4.4.

Consider the commutant of the Cartan subalgebra in
$\mathfrak{so}(4,\mathbb{C}),$ the roots decomposition will give us the
following basis
$\beta_{\mathfrak{so}^{*}(4,\mathbb{C})}=\left(h_{1},h_{2},\epsilon_{12}^{-},\epsilon_{12}^{+},\epsilon_{21}^{-},\hat{\epsilon}_{12}^{+}\right).$
Under the basis $\beta_{\mathfrak{so}^{*}(4)},$ all the indecomposable
elements in the symmetric centralizer $S\left(D_{2}\right)^{\mathfrak{h}}$ are

$\displaystyle A_{1}=h_{1},\text{ }A_{2}=h_{2}\in\mathcal{Q}_{1};\text{
}\quad\text{ }\epsilon_{12}^{-}\epsilon_{21}^{-},\text{
}\epsilon_{12}^{+}\hat{\epsilon}_{12}^{+}\in\mathcal{Q}_{2},$

which generate a commutative Poisson algebra
$\mathcal{Q}_{D_{2}}(0)=\mathfrak{h}^{*}\oplus\mathcal{Q}_{2}.$ This implies
that $\mathcal{Q}_{D_{2}}(0)$ is isomorphic to a polynomial ring
$\mathbb{C}\left[A_{1},A_{2},\epsilon_{12}^{-}\epsilon_{21}^{-},\epsilon_{12}^{+}\hat{\epsilon}_{12}^{+}\right].$
It is finitely-generated, and forms an integrable system with the families of
Hamiltonian $\mathcal{H}=\alpha h_{1}+\beta h_{2}+\gamma h_{1}h_{2}.$ Here
$\alpha,\beta$ and $\gamma\in\mathbb{R}.$

###  4.3. The explicit generators of $U(C_{n})^{\mathfrak{h}}$

For a sympletic Lie algebra $\mathfrak{sp}(2n,\mathbb{C}),$ its Cartan
subalgebra is defined by
$\mathfrak{h}=\left\\{H\in\mathfrak{sp}(2n,\mathbb{C}):H=\sum_{j=1}^{n}a_{j}\left(E_{j}-E_{n+j,n+j}\right),\text{
}h_{j}(H)=a_{j},\text{ }h_{j}\in\mathfrak{h}^{*}\right\\}.$ The positive roots
are given by $\alpha_{j,k}^{\pm}=\alpha_{j}\pm\alpha_{k},\text{ for all }1\leq
j<k\leq n,$ together with $\alpha_{i,i}=\alpha_{i},$ for all $1\leq i\leq n,$
where $\epsilon_{i},\epsilon_{j}$ are linear functional on $\mathfrak{h}^{*},$
which are generated by simple roots
$\Delta_{C_{n}}=\\{\alpha_{12}^{-},\ldots,\alpha_{n-1n}^{-},2\alpha_{n}\\}.$
Moreover, the corresponding between $\mathfrak{g}_{\alpha_{jk}}$ to their
corresponding roots are

$\displaystyle\begin{matrix}2\alpha_{j}\rightarrow E_{j,n+j}&1\leq j\leq n\\\
-2\alpha_{j}\rightarrow E_{n+j,j}&1\leq j\leq n\\\
\alpha_{j}-\alpha_{k}\rightarrow E_{j,k}-E_{n+j,n+k}&1\leq j<k\leq n\\\
\alpha_{j}+\alpha_{k}\rightarrow E_{j,k+n}-E_{k,j+n}&1\leq j<k\leq n\\\
-(\alpha_{j}+\alpha_{k})\rightarrow E_{j+n,k}-E_{n+k,j}&1\leq j<k\leq
n\end{matrix}$

In particular, the roots in $\Phi_{C_{n}}$ satisfy the following decomposition

(52) $\displaystyle
R\left(\epsilon_{jk}^{-}\right):=\alpha_{jk}^{-}=\sum_{l=j}^{k-1}\alpha_{ll+1},\quad
R\left(\epsilon_{j}\right):=\alpha_{j}=2\sum_{l=j}^{n-1}\alpha_{ll+1}+2\alpha_{n},\quad
R\left(\epsilon_{jk}^{+}\right):=\alpha_{jk}^{+}=\alpha_{jk}^{-}+\alpha_{k},$

with $1\leq j<k\leq n.$ By a simply rewriting of the root in
$\eqref{eq:root2},$ we have that
$\alpha_{jk}^{+}=\alpha_{jn}^{-}+\alpha_{kn}^{-}+2\alpha_{n}.$ For all
$h_{i}\in\mathfrak{h}^{*},$ the $ad^{*}(h_{i})$-action on the roots will
generate the following diagonalizable terms

$\displaystyle\\{h_{i},\epsilon_{jk}^{-}\\}=\mu_{j,k}^{i}\epsilon_{jk}^{-}\text{
and
}\\{h_{i},\epsilon_{j}\\}=\left(2\mu_{j,n}^{i}+\mu_{n}^{i}\right)\epsilon_{j}=\mu_{j,j}^{i}\epsilon_{j},$

where

(53) $\displaystyle\mu_{j,j+1}^{j\pm
1}=-1,\quad\mu_{j,j+1}^{j}=2,\quad\mu_{n-1,n-1}^{n}=-2,\quad\mu_{j,j+1}^{l}=0,\text{
}j\neq l\pm 1,l=1,\ldots,n.$

We now focus on the classification of explicit generators for the Poisson
algebra $\mathcal{Q}_{C_{n}}(d^{\prime\prime})$. Similarly to the construction
in Section 4.1, the monomials in Cartan commutant can be again divided into
five classes. The relation between those root systems implies that the
monomial $p(\textbf{x}_{C})$ in terms of permutation root vectors or
$\mathbb{Z}_{2}^{L}$-span will have the same form as the monomials in $(a)$
and $(b)$, as before. From the root system in $\eqref{eq:root2}$, the elements
in the Cartan commutant $U\left(C_{n}\right)^{\mathfrak{h}}$ should have a
similar form in $\eqref{eq:41}.$ Furthermore, the roots corresponding to
$p(\textbf{x}_{C})\in\mathcal{Q}_{C_{n}}(d^{\prime\prime})$ have the form of

(54)
$\displaystyle\sum_{s}^{r}\alpha_{i_{s}j_{s}}^{-}+\sum_{w=1}^{\mu}\left(\alpha_{k_{w}n}^{-}+\alpha_{l_{w}n}^{-}\right)+\sum_{w=\mu+1}^{t}\left(\alpha_{nk_{w}}^{-}+\alpha_{nl_{w}}^{-}\right)+2\left(\sum_{a=1}^{\nu}\alpha_{s_{a}n}^{-}\right)+2\left(\sum_{b=\nu+1}^{q}\alpha_{ns_{b}}^{-}\right)=0$

with $\mu,\nu\in\mathbb{N}_{0}.$ We can show that the indecomposable monomials
in $\mathcal{Q}_{C_{n}}(d^{\prime\prime})$ have the following forms.

Define a mapping
$f:=(f_{1},f_{2},f_{3}):\Phi_{\mathfrak{so}(2n+1,\mathbb{C})|_{\mathfrak{g}_{+}}}\rightarrow\Phi_{\mathfrak{sp}(2n,\mathbb{C})|_{\mathfrak{g}_{+}}}$
given by
$(\alpha_{ij}^{-},\alpha_{kl}^{+},\alpha_{p})\mapsto(\alpha_{ij}^{-},\alpha_{kl}^{+},2\alpha_{p})$
from the root vectors to their corresponding roots, which establishes the
relation between $\Phi_{B_{n}}$ and $\Phi_{D_{n}}.$ It is clear that
$f_{1}=f_{2}=id$ and $f_{3}=id,$ where $id$ is the identity map. We can define
the same connection to negative roots. Hence, the permutation roots and long
roots are the same. We can then omit the classifications in the spaces
$\mathcal{Q}_{C_{n}}(d^{\prime\prime})\cap S_{n}$-span,
$\mathcal{Q}_{C_{n}}(d^{\prime\prime})\cap\mathbb{Z}_{2}^{L}$-span,
$\mathcal{Q}_{C_{n}}(d^{\prime\prime})\cap\mathbb{Z}_{2}^{S}$-span and
$\mathcal{SK}_{1}.$

(c) Let
$\mathcal{SK}_{2}:=\left(S_{n}\text{-span}\otimes\mathbb{Z}_{2}^{S}\text{-span}\right)$
$\cap\mathcal{Q}_{C_{n}}(d^{\prime\prime})$. Then there exists a
$\prod_{s=1}^{r}\left[\epsilon_{i_{s}j_{s}}^{-}\right]\in S_{n}$-span such
that
$p=\prod_{s=1}^{r}\left[\epsilon_{i_{s}j_{s}}^{-}\right]m_{2}(\textbf{x}_{S})\in\mathcal{SK}_{2}$
is indecomposable if and only if $\deg m_{2}=2.$ In particular,
$\mathcal{SK}_{2}$ is spanned by

(55)
$\displaystyle\left(\left[\epsilon_{ij}^{-}\right]\right)^{2}\hat{\epsilon}_{i}\epsilon_{j},\text{
and
}\left(\left[\epsilon_{ji}^{-}\right]\right)^{2}\epsilon_{i}\hat{\epsilon}_{j}$

with $1\leq i\neq j\leq n.$ This also provides the maximum degree of
indecomposable monomials $2n.$

(d) Let
$\mathcal{K}_{1}\mathcal{K}_{2}:=\left(\mathbb{Z}_{2}^{S}\text{-span}\otimes\mathbb{Z}_{2}^{L}\text{-span}\right)\cap\mathcal{Q}_{C_{n}}(d^{\prime\prime})$.
Suppose that

$\displaystyle m_{1}(\textbf{x}_{L_{1}})=\epsilon_{kl}^{+}\prod_{1\leq
w\leq\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{\mu+1\leq u\leq
2\mu}\hat{\epsilon}_{k_{u}l_{u}}^{+}\text{ and
}m_{1}(\textbf{x}_{L_{2}})=\prod_{1\leq
w\leq\mu}\epsilon_{k_{w}l_{w}}^{+}\prod_{\mu+1\leq u\leq
2\mu}\hat{\epsilon}_{k_{u}l_{u}}^{+}\in\mathbb{Z}_{2}\text{-span}.$

Then

(56) $\displaystyle
m\big{(}\textbf{x}_{L_{1}},\hat{\epsilon}_{s}\big{)}=m_{1}(\textbf{x}_{L_{1}})\hat{\epsilon}_{s},\quad\text{
}m\big{(}\textbf{x}_{L_{2}},\epsilon_{a},\hat{\epsilon}_{b}\big{)}=\left(m_{1}(\textbf{x}_{L_{2}})\right)^{2}\epsilon_{s_{a}}\hat{\epsilon}_{s_{b}}\in\mathcal{K}_{1}\mathcal{K}_{2}.$

are indecomposable.

(e) Suppose that there exists a $\left[g(\textbf{x}_{1})\right]\in S_{n}$-span
with $n\geq 3$ such that
$g(\textbf{x}_{1})m_{1}(\textbf{x}_{L})m_{2}(\textbf{x}_{S})\in\mathcal{Q}_{C_{n}}(d^{\prime\prime})$
is indecomposable with none of $g(\textbf{x}_{1})$ $m_{1}(\textbf{x}_{L})$ and
$m_{2}(\textbf{x}_{S})$ equal to $1$. Then $\deg m_{2}(\textbf{x}_{S})\leq 2.$
Conversely, if $\deg m_{2}(\textbf{x}_{S})\leq 2,$ then there exist
$\left[g(\textbf{x}_{I})\right],\left[g(\textbf{x}_{II})\right]\in S_{n}$-span
such that

$\displaystyle
p_{I}\left(\textbf{x}_{I},\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{2\mu}l_{2\mu}}^{+},\epsilon_{a},\hat{\epsilon}_{b}\right)=$
$\displaystyle\left[g(\textbf{x}_{I})\right]\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{l_{w}l_{u}}^{+}\epsilon_{a}\hat{\epsilon}_{b}\text{
with }a\neq b;$ $\displaystyle
p_{II}\left(\textbf{x}_{II},\epsilon_{kl}^{+},\epsilon_{k_{1}l_{1}}^{+},\ldots,\hat{\epsilon}_{k_{2\mu}l_{2\mu}}^{+},\hat{\epsilon}_{a}\right)=$
$\displaystyle\left[g(\textbf{x}_{II})\right]\epsilon_{kl}^{+}\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\hat{\epsilon}_{a};$

are indecomposable monomials. Here $\textbf{x}_{I}$ and $\textbf{x}_{II}$ are
distinct permutation types of root vectors.

With the information above, we conclude that
$\mathcal{Q}_{C_{n}}(2n-1)=\mathbb{C}\langle\textbf{Q}_{2n}\rangle$ defines a
polynomial algebra.

##  5\. Non-commutative polynomial algebras in universal enveloping algebras
with rank $3$ and its embedding chain

In this section, we will provide the classification of the generators for
Cartan centralizers of complex semisimple Lie algebras for a certain rank.
Notice that the classification for Lie algebras with the rank less than $2$ is
straightforward. As an example, we present a case with the rank of
$\mathfrak{g}$ equals to $2$ as follows:

######  Example 5.1.

Notice that we will omit the case if $n=1.$ Starting from $n=2.$ In $B_{2},$
we have
$\Phi_{B_{2}}=\left\\{\alpha_{12}^{-},\alpha_{21}^{-},\alpha_{12}^{+},\hat{\alpha}_{12}^{+},\alpha_{1},\hat{\alpha}_{1},\alpha_{2},\hat{\alpha}_{2}\right\\}$
with a set of simple root
$\Delta_{B_{2}}=\left\\{\alpha_{12}^{-},\alpha_{2}\right\\}.$ Then the
indecomposable polynomial solutions of the systems of PDEs
$\\{p,h_{i}\\}_{o}=0$ are all the Cartan elements and

$\displaystyle\epsilon_{12}^{-}\epsilon_{21}^{-},\quad\epsilon_{12}^{+}\hat{\epsilon}_{12}^{+},\quad\epsilon_{1}\hat{\epsilon}_{1},\quad\epsilon_{2}\hat{\epsilon}_{2}\in\mathcal{Q}_{2}$
$\displaystyle\epsilon_{12}^{-}\epsilon_{2}\hat{\epsilon}_{1},\text{
}\epsilon_{21}^{-}\epsilon_{1}\hat{\epsilon}_{2},\text{
}\epsilon_{12}^{+}\hat{\epsilon}_{1}\hat{\epsilon}_{2},\text{
}\hat{\epsilon}_{12}^{+}\epsilon_{1}\epsilon_{2}\in\mathcal{Q}_{3}$
$\displaystyle\epsilon_{12}^{-}\epsilon_{12}^{+}\left(\hat{\epsilon}_{1}\right)^{2},\text{
}\epsilon_{12}^{-}\hat{\epsilon}_{12}^{+}\left(\epsilon_{2}\right)^{2},\text{
}\epsilon_{21}^{-}\epsilon_{12}^{+}\left(\hat{\epsilon}_{2}\right)^{2},\text{
}\epsilon_{21}^{-}\hat{\epsilon}_{12}^{+}\left(\epsilon_{1}\right)^{2}\in\mathcal{Q}_{4}.$

They form a cubic algebra $\mathcal{Q}_{B_{2}}(3)$ with
$\dim_{FL}\mathcal{Q}_{B_{2}}(3)=14.$

In what follows, we will have a look at Lie algebras with the rank $3.$ That
is, $\mathfrak{g}=\mathfrak{sl}(4,\mathbb{C})\cong A_{3},$
$\mathfrak{so}(6,\mathbb{C})\cong D_{3}$, $\mathfrak{so}(7,\mathbb{C})\cong
B_{3}$ and $\mathfrak{sp}(6,\mathbb{C})\cong C_{3}$. Notice that the
polynomial algebra $\mathcal{Q}_{A_{3}}(3)$ has been fully studied in [6], for
which reason we omit that case in this paper. In the following, we will
formulate explicit expressions for all polynomial generators in
type-$B_{3},D_{3},C_{3}$ Cartan commutants using the classification results
listed in Section 4. We emphasize the fact that, although $A_{3}$ and $D_{3}$
are isomorphic, the analysis of $D_{3}$ is itself of interest, in order to
illustrate the structure of the polynomial algebra for higher rank.

Instead of using the coordinate argument to express indecomposable monomials,
we now introduce a representative formula via the pair of indices
$(i_{s},j_{s}),(k_{w},l_{w})$, $(k_{u},l_{u})$, $s_{a}$ and $s_{b}$ in root
vectors. That is, the generic formula of indecomposable monomials can be
rewritten in the indices argument as follows:

(57) $\displaystyle
p_{[i_{1}j_{1}^{-}],\ldots,[i_{r}j_{r}^{-}];k_{1}l_{1}^{+},\ldots,\hat{k_{t}l_{t}}^{+};s_{1},\ldots,s_{\nu},\hat{s}_{1},\ldots,\hat{s}_{q}}.$

Here $[i_{1}j_{1}^{-}],\ldots,[i_{r}j_{r}^{-}]$ represents all the possible
choices of permutation root vectors, where
$k_{1}l_{1}^{+},\ldots,\hat{k_{t}l_{t}}^{+}$ and $s_{1},\ldots,s_{\nu},$
$\hat{s}_{1},\ldots,\hat{s}_{q}$ represent long and short root vectors
respectively. Note that the double comma ; in $\eqref{eq:indrep}$ divides each
type of root vectors. Notice that $\eqref{eq:indrep}$ is well-defined as the
ordering of indices in each section does not change the form of $p.$ For
instance, in [5], it is shown that one of the generators in Cartan commutants
of $B_{4}$-type are

$\displaystyle\begin{matrix}p_{[4i^{-}],[l4^{-}];ij^{+},\hat{kl}^{+};\hat{j},\hat{k}},\quad\quad
p_{[i4^{-}],[l4^{-}];\hat{ij}^{+},kl^{+};j,k}\\\
p_{[k4^{-}],[4j^{-}];ij^{+},\hat{kl}^{+};\hat{i},\hat{l}},\quad\quad
p_{[4k^{-}],[j4^{-}];\hat{ij}^{+},kl^{+};i,l}\end{matrix}\in\mathcal{Q}_{5},\mathcal{Q}_{6},\mathcal{Q}_{7}$

where $1\leq i\neq j\neq k\neq l\leq 4.$ Here
$p_{[4i^{-}],[l4^{-}];ij^{+},\hat{kl}^{+};\hat{j},\hat{k}}=\left[\epsilon_{4i}^{-}\right]\left[\epsilon_{l4}^{-}\right]\epsilon_{ij}^{+}\hat{\epsilon}_{kl}^{+}\hat{\epsilon}_{j}\hat{\epsilon}_{k}.$
In what follows, once we deduce the explicit formula, we will use the indices
argument to express the indecomposable monomials.

###  5.1. Explicit generators for $\mathcal{Q}_{B_{3}}(d)$

Starting with a Lie algebra $\mathfrak{so}(7,\mathbb{C})$, under an ordered
dual basis
$\beta_{\mathfrak{so}^{*}(7,\mathbb{C})}=\left\\{h_{1},h_{2},h_{3},\epsilon_{ij}^{-},\epsilon_{ji}^{-},\epsilon_{kl}^{+},\hat{\epsilon}_{kl}^{+},\epsilon_{a},\hat{\epsilon}_{b}:1\leq
i,j,k,l,a,b\leq 3\right\\},$ we are able to provide a classification of Cartan
generators from statements (a) to (e) in Section 4.1. By definition,
$\mathfrak{h}^{*}=\mathrm{span}\left\\{h_{1},h_{2},h_{3}\right\\}$ forms the
first layer in $\mathcal{Q}_{B_{3}}(d).$ We then provide the explicit
construction for the rest of the layers.

From observation (a), $p\in\mathcal{Q}_{B_{3}}(d)$ has the form of $g$,
$m_{1}$ or $m_{2}$. In detail, we have

$\displaystyle p_{12^{-},21^{-}}=\epsilon_{12}^{-}\epsilon_{21}^{-},\quad
p_{[13^{-}],31^{-}}=\left[\epsilon_{13}^{-}\right]\epsilon_{31}^{-},\quad
p_{[23^{-}],32^{-}}=\left[\epsilon_{23}^{-}\right]\epsilon_{32}^{-}\in\mathcal{Q}_{A_{2}}(2),$
$\displaystyle
p_{12^{+},\hat{12}^{+}}=\epsilon_{12}^{+}\hat{\epsilon}_{12}^{+},\quad
p_{13^{+},\hat{13}^{+}}=\epsilon_{13}^{+}\hat{\epsilon}_{13}^{+},\quad
p_{23^{+},\hat{23}^{+}}=\epsilon_{23}^{+}\hat{\epsilon}_{23}^{+}.$

and $p_{a,\hat{a}}=\epsilon_{a}\hat{\epsilon}_{a}$ for all $1\leq a\leq 3,$
where
$\left[\epsilon_{13}^{-}\right]:=\left\\{\epsilon_{13}^{-},\epsilon_{12}^{-}\epsilon_{23}^{-}\right\\}$
and
$\left[\epsilon_{23}^{-}\right]:=\left\\{\epsilon_{23}^{-},\epsilon_{21}^{-}\epsilon_{13}^{-}\right\\}.$
They generate the subspace $\mathcal{Q}_{2}.$

In observation (b), the monomial has the form of $gm_{1}\in\mathcal{SK}_{1}$.
Since $\alpha_{n}\notin R(m_{1})$, we have that $\deg m_{1}\in 2\mathbb{N}.$
Due to the distinct length of the long roots in the pair
$\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}$, under the
constraint $\alpha_{n}\notin R(m_{1})$, the monomial $m_{1}$ has been
classified into three forms as follows:

$\displaystyle\text{ (A1)
}\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\text{
with $1\leq k_{w},l_{w}\neq k_{u},l_{u}\leq 3$};$ (58) $\displaystyle\text{
(A2)
}\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}k_{u}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\text{
with $1\leq k_{w}\neq l_{u}\leq 3$};$ $\displaystyle\text{ (A3)
}\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\prod_{1\leq
a\neq b\neq c\leq 3}\epsilon_{ab}^{+}\hat{\epsilon}_{bc}^{+}\text{ with $1\leq
k_{w},l_{w}\neq k_{u},l_{u}\leq 3$}.$

Notice that if the case (A1) and (A3) holds, then $n\geq 4.$ Therefore the
only possible form of $m_{1}$ is the case (A2) in $\eqref{eq:classif}.$ Then
all the possible forms of $m_{1}$ are given by

(59)
$\displaystyle\begin{matrix}m_{1}^{(a)}(\epsilon_{12}^{+},\hat{\epsilon}_{23}^{+})=\epsilon_{12}^{+}\hat{\epsilon}_{23}^{+},\quad
m_{1}^{(b)}(\epsilon_{12}^{+},\hat{\epsilon}_{13}^{+})=\epsilon_{12}^{+}\hat{\epsilon}_{13}^{+},\quad
m_{1}^{(c)}(\epsilon_{13}^{+},\hat{\epsilon}_{23}^{+})=\epsilon_{13}^{+}\hat{\epsilon}_{23}^{+};\\\
\hat{m}_{1}^{(a)}(\hat{\epsilon}_{12}^{+},\epsilon_{23}^{+})=\hat{\epsilon}_{12}^{+}\epsilon_{23}^{+},\quad\hat{m}_{1}^{(b)}(\hat{\epsilon}_{12}^{+},\epsilon_{13}^{+})=\hat{\epsilon}_{12}^{+}\epsilon_{13}^{+},\quad\hat{m}_{1}^{(c)}(\hat{\epsilon}_{13}^{+},\epsilon_{23}^{+})=\hat{\epsilon}_{13}^{+}\epsilon_{23}^{+}.\end{matrix}$

Then

(60)
$\displaystyle\begin{matrix}p_{[13^{-}];12^{+},\hat{23}^{+}}=\left[\epsilon_{13}^{-}\right]m_{1}^{(a)},\quad
p_{[23^{-}];12^{+},\hat{13}^{+}}=\left[\epsilon_{23}^{-}\right]m_{1}^{(b)},\quad
p_{[12^{-}];13^{+},\hat{23}^{+}}=\left[\epsilon_{12}^{-}\right]m_{1}^{(c)},\\\
p_{[31^{-}];23^{+},\hat{12}^{+}}=\left[\epsilon_{31}^{-}\right]\hat{m}_{1}^{(a)},\quad
p_{[32^{-}];13^{+},\hat{12}^{+}}=\left[\epsilon_{32}^{-}\right]\hat{m}_{1}^{(b)},\quad
p_{[21^{-}];23^{+},\hat{13}^{+}}=\left[\epsilon_{21}^{-}\right]\hat{m}_{1}^{(c)},\end{matrix}$

are indecomposable monomials in either $\mathcal{Q}_{3}$ and
$\mathcal{Q}_{4},$ where

(61)
$\displaystyle\begin{matrix}\left[\epsilon_{13}^{-}\right]:=\left\\{\epsilon_{13}^{-},\epsilon_{12}^{-}\epsilon_{23}^{-}\right\\},\text{
}\left[\epsilon_{23}^{-}\right]:=\left\\{\epsilon_{23}^{-},\epsilon_{21}^{-}\epsilon_{13}^{-}\right\\}\text{
and
}\left[\epsilon_{12}^{-}\right]:=\left\\{\epsilon_{12}^{-},\epsilon_{13}^{-}\epsilon_{32}^{-}\right\\};\\\
\left[\epsilon_{31}^{-}\right]:=\left\\{\epsilon_{31}^{-},\epsilon_{32}^{-}\epsilon_{21}^{-}\right\\},\text{
}\left[\epsilon_{32}^{-}\right]:=\left\\{\epsilon_{32}^{-},\epsilon_{31}^{-}\epsilon_{12}^{-}\right\\}\text{
and
}\left[\epsilon_{21}^{-}\right]:=\left\\{\epsilon_{21}^{-},\epsilon_{23}^{-}\epsilon_{31}^{-}\right\\}.\end{matrix}$

We now look at the monomial from observation (c). The monomial lives in the
subspace $\mathcal{SK}_{2}$ with the form of $[g]\prod_{1\leq a,b\leq
3}\epsilon_{s_{a}}\hat{\epsilon}_{s_{b}}.$ Using $\eqref{eq:id5770}$, the
monomials span $\mathcal{SK}_{2}$ are

(62)
$\displaystyle\left.\begin{matrix}p_{[12^{-}],\hat{1},2}=\left[\epsilon_{12}^{-}\right]\hat{\epsilon}_{1}\epsilon_{2},\quad
p_{[21^{-}],1,\hat{2}}=\left[\epsilon_{21}^{-}\right]\epsilon_{1}\hat{\epsilon}_{2}\\\
p_{[13^{-}],\hat{1},3}=\left[\epsilon_{13}^{-}\right]\hat{\epsilon}_{1}\epsilon_{3},\quad
p_{[31^{-}],1,\hat{3}}=\left[\epsilon_{31}^{-}\right]\epsilon_{1}\hat{\epsilon}_{3}\\\
p_{[23^{-}],\hat{2},3}=\left[\epsilon_{23}^{-}\right]\hat{\epsilon}_{2}\epsilon_{3},\quad
p_{[32^{-}],2,\hat{3}}=\left[\epsilon_{32}^{-}\right]\epsilon_{2}\hat{\epsilon}_{3}\end{matrix}\right\\}\in\mathcal{Q}_{3}\text{
or }\mathcal{Q}_{4}.$

Next, the monomials that span the subspace $\mathcal{K}_{1}\mathcal{K}_{2}$
are classified into two types: $\alpha_{n}\notin R\left(m_{1}\right)$ and
$\alpha_{n}\in R\left(m_{1}\right).$ If $\alpha_{n}\notin
R\left(m_{1}\right)$, then $\alpha_{n}\notin R\left(m_{2}\right)$ and $m_{1}$
have the form of $\eqref{eq:classif}.$ From the argument in observation (b),
all the possible forms of $m_{1}$ are given by $\eqref{eq:m1}$. Then the
indecomposable monomials should have the form of

(63)
$\displaystyle\left.\begin{matrix}p_{12^{+},\hat{23}^{+};\hat{1},3}=m_{1}^{(a)}\hat{\epsilon}_{1}\epsilon_{3},\quad
p_{23^{+},\hat{12}^{+};1,\hat{3}}=\hat{m}_{1}^{(a)}\epsilon_{1}\hat{\epsilon}_{3}\\\
p_{12^{+},\hat{13}^{+};\hat{2},3}=m_{1}^{(b)}\hat{\epsilon}_{2}\epsilon_{3},\quad
p_{13^{+},\hat{12}^{+};2,\hat{3}}=\hat{m}_{1}^{(b)}\epsilon_{2}\hat{\epsilon}_{3}\\\
p_{13^{+},\hat{23}^{+};\hat{1},2}=m_{1}^{(c)}\hat{\epsilon}_{1}\epsilon_{2},\quad
p_{13^{+},\hat{23}^{+};1,\hat{3}}=\hat{m}_{1}^{(c)}\epsilon_{1}\hat{\epsilon}_{2}.\end{matrix}\right\\}\in\mathcal{Q}_{4}.$

On the other hands, if $\alpha_{n}\in R\left(m_{1}\right),$ we must have that
$\alpha_{n}\in R\left(m_{2}\right)$. Based on the results in
$\eqref{eq:classif}$, $m_{1}$ can only admit one of the following forms

$\displaystyle\text{ (B1) }\prod_{w=1}^{t}\epsilon_{k_{w}l_{w}}^{+}\text{ with
$1\leq k_{w}\neq l_{w}\leq 3$};$ $\displaystyle\text{ (B2) }\prod_{1\leq
i,j\leq
3}\epsilon_{ij}^{+}\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\text{
with $1\leq k_{w},l_{w}\leq 3$};$ (64) $\displaystyle\text{ (B3) }\prod_{1\leq
i,j\leq
3}\epsilon_{ij}^{+}\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}k_{u}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\text{
with $1\leq k_{w}\neq l_{u}\leq 3$};$ $\displaystyle\text{ (B4) }\prod_{1\leq
i,j\leq
3}\epsilon_{ij}^{+}\prod_{w=1}^{\mu}\prod_{u=\mu+1}^{2\mu}\epsilon_{k_{w}l_{w}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\prod_{1\leq
a,b,c\leq n}\epsilon_{ab}^{+}\hat{\epsilon}_{bc}^{+}\text{ with $1\leq
k_{w},l_{w}\neq k_{u},l_{u}\leq 3$}.$

Here $1\leq i,j,k_{w},l_{w},k_{u},l_{u},a,b,c\leq 3.$ From the discussion
above, we can consider that only (B1) and (B3) as the form of $m_{1}$.
Starting with $m_{1}$ is of the form (B1). Since $|R(m_{1})|=4$, the
indecomposable monomials in $\mathcal{SK}_{2}$ have the form of
$m_{1}\hat{\epsilon}_{s_{1}}\hat{\epsilon}_{s_{2}}$ with $1\leq s_{1}\neq
s_{2}\leq 3$. Therefore, all the possible choice of indecomposable monomials
are

(65)
$\displaystyle\left.\begin{matrix}p_{12^{+};\hat{1},\hat{2}}=\epsilon_{12}^{+}\hat{\epsilon}_{1}\hat{\epsilon}_{2},\quad
p_{13^{+};\hat{1},\hat{3}}=\epsilon_{13}^{+}\hat{\epsilon}_{1}\hat{\epsilon}_{3},\quad
p_{23^{+};\hat{2},\hat{3}}=\epsilon_{23}^{+}\hat{\epsilon}_{2}\hat{\epsilon}_{3};\\\
p_{\hat{12}^{+};1,2}=\hat{\epsilon}_{12}^{+}\epsilon_{1}\epsilon_{2},\quad
p_{\hat{13}^{+};1,3}=\hat{\epsilon}_{13}^{+}\epsilon_{1}\epsilon_{3},\quad
p_{\hat{23}^{+};2,3}=\hat{\epsilon}_{23}^{+}\epsilon_{2}\epsilon_{3}\end{matrix}\right\\}\in\mathcal{Q}_{3}$

Moreover, if the monomial $m_{1}$ has the form of (B2) in
$\eqref{eq:classif2}$, then the indecomposable polynomials in
$\mathcal{Q}_{5}$ have the form

$\displaystyle\begin{matrix}p_{13^{+},12^{+},\hat{23}^{+};\hat{1}^{2}}=\epsilon_{13}^{+}m_{1}^{(a)}\hat{\epsilon}_{1}^{2},\quad
p_{23^{+},12^{+},\hat{13}^{+};\hat{2}^{2}}=\epsilon_{23}^{+}m_{1}^{(b)}\hat{\epsilon}_{2}^{2},\quad
p_{12^{+},13^{+},\hat{23}^{+};\hat{1}^{2}}=\epsilon_{12}^{+}m_{1}^{(c)}\hat{\epsilon}_{1}^{2},\\\
p_{\hat{13}^{+},23^{+},\hat{12}^{+};1^{2}}=\hat{\epsilon}_{13}^{+}\hat{m}_{1}^{(a)}\epsilon_{1}^{2},\quad
p_{\hat{23}^{+},13^{+},\hat{12}^{+};2^{2}}=\hat{\epsilon}_{23}^{+}\hat{m}_{1}^{(b)}\epsilon_{2}^{2},\quad
p_{\hat{12}^{+},23^{+},\hat{13}^{+};1^{2}}=\hat{\epsilon}_{12}^{+}\hat{m}_{1}^{(c)}\epsilon_{1}^{2}.\end{matrix}$

Finally, we will list all the indecomposable monomials from observation (e).
From the discussion above, with different choices of $m_{1}$, the
indecomposable monomials have the following forms

$\displaystyle p_{I}=$ $\displaystyle\text{
}\left[g_{I}\right]\epsilon_{k_{w}k_{u}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\epsilon_{a}\hat{\epsilon}_{b}\text{
with $1\leq k_{w}\neq l_{u},a\neq b\leq 3$};$ $\displaystyle p_{II}=$
$\displaystyle\text{
}\left[g_{II}\right]\epsilon_{kl}^{+}\hat{\epsilon}_{s_{1}}\hat{\epsilon}_{s_{2}}\text{
with $1\leq k,l\leq 3$};$ $\displaystyle p_{II}^{\prime}=$
$\displaystyle\text{
}\left[g_{II}^{\prime}\right]\epsilon_{kl}^{+}\epsilon_{k_{w}k_{u}}^{+}\
\hat{\epsilon}_{k_{u}l_{u}}^{+}\hat{\epsilon}_{s_{1}}\hat{\epsilon}_{s_{2}}\text{
with $1\leq k_{w}\neq l_{u},k,l\leq 3$.}$

We now show that $p_{II}^{\prime}\in\mathcal{Q}_{B_{n}}(d)$ are decomposable
if $s_{1}\neq s_{2}$. Suppose the contrary. That is, there exists a
$p_{II}^{\prime}\in\mathcal{Q}_{B_{n}}(d)$ with $s_{1}\neq s_{2}$ such that
$p_{II}^{\prime}$ is indecomposable. Since $R\left(p_{II}^{\prime}\right)=0,$
it follows that

$R(g_{II}^{\prime})=-\left(\alpha_{k3}^{-}+\alpha_{l3}^{-}+\alpha_{k_{w}3}^{-}+\alpha_{3l_{u}}^{-}+\alpha_{3s_{1}}^{-}+\alpha_{3s_{2}}^{-}\right).$

Then all the possible combinations of positive and negative roots in
$R\left(g_{II}^{\prime}\right)$ are given by

| $(k_{w}3)$ | $(k3)$ | $(l3)$
---|---|---|---
$(3l_{u})$ | $(k_{w}l_{u})$ | $(kl_{w})$ | $(ll_{u})$
$(3s_{1})$ | $(k_{w}s_{1})$ | $(ks_{1})$ | $(ls_{1})$
$(3s_{2})$ | $(k_{w}s_{2})$ | $(ks_{2})$ | $(ls_{2})$

Table 5.1

Here we use index pairs to represent the positive and negative roots in each
entry of Table 5.1. For instance, using the pair $(k_{w}l_{u}),$ we can
replace the identity
$\alpha_{k_{w}3}^{-}+\alpha_{3l_{u}}^{-}=\alpha_{k_{w}l_{u}}^{-}$ in the first
entry. Then it is sufficient to write Table 5.1 as a non-singular matrix

(66) $\displaystyle R=\begin{pmatrix}(k_{w}l_{u})&(kl_{w})&(ll_{u})\\\
(k_{w}s_{1})&(ks_{1})&(ls_{1})\\\ (k_{w}s_{2})&(ks_{2})&(ls_{2})\end{pmatrix}$

such that all the possible combinations of pairs forming $g_{II}^{\prime}$ can
be taken from each column or row of $R.$ Since $R$ is non-singular, the number
of choices of the indices to form $R(g_{II}^{\prime})$ equals to the rank of
$R.$ In particular, there are $6$ distinct choices and all the pairs induce
that indecomposable monomials are given by

(67)
$\displaystyle(kl_{u})(k_{w}s_{1})(ls_{2}),\quad(kl_{u})(ls_{1})(k_{w}s_{2}),\quad(ll_{u})(ks_{1})(k_{w}s_{2}),\quad(ll_{u})(k_{w}s_{1})(ks_{2}).$

Since $\mathrm{rank}\mathfrak{g}=3,$ we must have that $s_{1}\neq s_{2},k\neq
l\in\\{k_{w},k_{u},l_{u}\\}\subset I_{3}:=\\{1,2,3\\}$ from the index pairs in
$p_{II}^{\prime}$. Notice that we must have that either $p_{i}=k$ or $l$ for
all $i.$ Otherwise, if $s_{1}=k$ and $s_{2}=l$ hold, by the identity
$\eqref{eq:60},$ we observe that $p_{II}^{\prime}$ is decomposable. Without
loss of generality, take $s_{1}=l.$ Then $\eqref{eq:ind}$ becomes

(68)
$\displaystyle(kl_{u})(k_{w}l)(ls_{2})\mathrel{\overset{S_{n}\text{-span}}{\scalebox{1.5}[1.0]{$\sim$}}}(kl_{u})(k_{w}s_{2})\mathrel{\overset{S_{n}\text{-span}}{\scalebox{1.5}[1.0]{$\sim$}}}(kl)(ll_{u})(k_{w}s_{2}),\quad(k_{w}l)(ll_{u})(ks_{2}).$

Hence there are two choices of indices: $(kl_{u})(k_{w}s_{2})$ and
$(k_{w}l)(ll_{u})(ks_{2})\mathrel{\overset{S_{n}\text{-span}}{\scalebox{1.5}[1.0]{$\sim$}}}(k_{w}l_{u})(ks_{2})$
from $\eqref{eq:ind2}.$ In particular, only $(kl_{u})(k_{w}s_{2})$ makes
$P_{II}^{\prime}$ indecomposable. We now determine the value of the indices
$s_{2}.$ By definition, $s_{2}\neq l_{u}.$ Moreover, if
$s_{2}\notin\\{k,l_{u},k_{w}\\},$ then $m_{1}$ contains at least $3$ non-equal
indices, which leads to a contradiction to $\mathrm{rank}\,\mathfrak{g}=3.$
Hence $s_{2}\in\\{k,l_{u},k_{w}\\}.$ Note that $s_{2}\neq k,$ otherwise
$(k_{w}k)(kl_{u})\mathrel{\overset{S_{n}\text{-span}}{\scalebox{1.5}[1.0]{$\sim$}}}(k_{w}l_{u})$
will lead to $p_{II}^{\prime}$ decomposable. This makes the choice of $s_{2}$
to be $k_{w}.$ Therefore, the indecomposable monomial $p_{II}^{\prime}$ has
the form of

$\displaystyle
p_{l_{u}k^{-};kl^{+},k_{w}k_{u}^{+},\hat{k_{u}l_{u}}^{+};\hat{l},\hat{k}_{w}}=\epsilon_{l_{u}k}^{-}\epsilon_{kl}^{+}\epsilon_{k_{w}k_{u}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\hat{\epsilon}_{l}\hat{\epsilon}_{k_{w}}\text{
with }l_{u}\neq k\neq l,k_{w}\in I_{3}.$

Since $\mathrm{rank}\,\mathfrak{g}=3,$ we have $k_{w}\in\\{l_{u},k,l\\}$.
Notice that if $l$ equals to either $l_{u}$ or $k,$ $p_{II}^{\prime}$ defined
above are decomposable. That is,

$\displaystyle p_{II}^{\prime}=$ $\displaystyle\text{
}p_{l_{u}k^{-};kk_{u}^{+},\hat{k_{u}l_{u}}^{+}}p_{kl^{+};\hat{k},\hat{l}}\text{
if }k_{w}=k$ $\displaystyle p_{II}^{\prime}=$ $\displaystyle\text{
}p_{kk_{u}^{+},\hat{k_{u}l_{u}}^{+}}p_{k_{w}k^{-};kl^{+};\hat{l},\hat{l}_{u}}\text{
if }k_{w}=l.$

Hence, we must have $l=k_{w}\in I_{3}.$ Thus, the indecomposable monomials of
the form $p_{II}^{\prime}$ are given by

(69) $\displaystyle
p_{l_{u}k^{-};kl^{+},lk_{u}^{+},\hat{k_{u}l_{u}}^{+};\hat{l}^{2}}=\epsilon_{l_{u}k}^{-}\epsilon_{kl}^{+}\epsilon_{lk_{u}}^{+}\hat{\epsilon}_{k_{u}l_{u}}^{+}\left(\hat{\epsilon}_{l}\right)^{2},\quad\hat{p}_{II}^{\prime}\in\mathcal{Q}_{6}$

with $l_{u}\neq k\neq l,k_{u}\in I_{3}.$ Using $\eqref{eq:m1}$,
$p_{II}^{\prime}$ in $\eqref{eq:m159}$ has the following forms

(70)
$\displaystyle\begin{matrix}\epsilon_{32}^{-}\left(\epsilon_{12}^{+}\right)^{2}\hat{\epsilon}_{23}^{+}\left(\hat{\epsilon}_{1}\right)^{2},\text{
}\epsilon_{23}^{-}\left(\epsilon_{13}^{+}\right)^{2}\hat{\epsilon}_{23}^{+}\left(\hat{\epsilon}_{1}\right)^{2},\text{
}\epsilon_{31}^{-}\left(\epsilon_{12}^{+}\right)^{2}\hat{\epsilon}_{13}^{+}\left(\hat{\epsilon}_{2}\right)^{2},\\\
\text{
}\epsilon_{13}^{-}\left(\epsilon_{23}^{+}\right)^{2}\hat{\epsilon}_{13}^{+}\left(\hat{\epsilon}_{2}\right)^{2},\text{
}\epsilon_{21}^{-}\left(\epsilon_{31}^{+}\right)^{2}\hat{\epsilon}_{21}^{+}\left(\hat{\epsilon}_{3}\right)^{2},\text{
}\epsilon_{12}^{-}\left(\epsilon_{23}^{+}\right)^{2}\hat{\epsilon}_{13}^{+}\left(\hat{\epsilon}_{3}\right)^{2}.\end{matrix}$

Together with the $\hat{\cdot}$ action on $p_{II}^{\prime}$, we can conclude
that $\dim_{FL}\mathcal{Q}_{6}=12.$

We now list all indecomposable monomials with the forms $p_{I}$ and $p_{II}$.
Starting with $p=p_{I},$ all the possible pairs in $R(g_{I})$ are
$(al_{u})(bk_{w})$, $(k_{w}l_{u})$ and $(ab).$ It is clear that
$(al_{u})(bk_{w})$ with $a\neq b,k_{w}\neq l_{u}\in I_{3}$ will lead to
indecomposable monomials. Then, using $\eqref{eq:m1}$, we have

(71)
$\displaystyle\begin{matrix}p_{23^{-};12^{+},\hat{23}^{+};2,\hat{1}},\quad
p_{12^{-};12^{+},\hat{23}^{+};3,\hat{2}},\quad
p_{13^{-};12^{+},\hat{13}^{+};1,\hat{2}},\\\
p_{12^{-};12^{+},\hat{13}^{+};3,\hat{1}},\quad
p_{13^{-};13^{+},\hat{23}^{+};2,\hat{3}},\quad
p_{23^{-};13^{+},\hat{23}^{+};3,\hat{1}}\end{matrix}$ (72)
$\displaystyle\begin{matrix}p_{32^{-};\hat{12}^{+},23^{+};\hat{2},1},\quad
p_{21^{-};\hat{12}^{+},23^{+};\hat{3},2},\quad
p_{31^{-};\hat{12}^{+},13^{+};\hat{1},2},\\\
p_{21^{-};\hat{12}^{+},13^{+};\hat{3},1},\quad
p_{31^{-};\hat{13}^{+},23^{+};\hat{2},3},\quad
p_{32^{-};\hat{13}^{+},23^{+};\hat{3},1}\end{matrix}$

are indecomposable monomials in $\mathcal{Q}_{5}$. Here
$p_{23^{-};12^{+},\hat{23}^{+};2,\hat{1}}=\epsilon_{23}^{-}m_{1}^{(a)}\epsilon_{2}\hat{\epsilon}_{1}.$
Now, assume that indecomposable polynomials have the form of $p_{II}$. For any
$k,l\in I_{3},$

(73) $\displaystyle\begin{matrix}p_{[32^{-}];12^{+};\hat{1},\hat{3}},\quad
p_{[31^{-}];12^{+};\hat{2},\hat{3}},\quad
p_{[23^{-}];13^{+};\hat{1},\hat{2}},\quad
p_{[21^{-}];13^{+};\hat{2},\hat{3}},\quad
p_{[13^{-}];23^{+};\hat{1},\hat{2}},\quad
p_{[12^{-}];23^{+};\hat{1},\hat{3}}\\\ p_{[23^{-}],\hat{12}^{+};1,3},\quad
p_{[13^{-}];\hat{12}^{+};2,3},\quad p_{[32^{-}],\hat{13}^{+};1,2},\quad
p_{[12^{-}];\hat{13}^{+};2,3},\quad p_{[31^{-}],\hat{23}^{+};1,2},\quad
p_{[21^{-}]:\hat{23}^{+};1,3}.\end{matrix}$

are indecomposable monomials in $\mathcal{Q}_{4}$ and $\mathcal{Q}_{5}$.

In conclusion, we have
$\mathcal{Q}_{B_{3}}(d)=\mathfrak{h}\oplus\bigoplus_{j\geq
2}^{6}\mathcal{Q}_{j}$ with

$\displaystyle\dim_{FL}\mathcal{Q}_{1}=$ $\displaystyle\text{
}\dim\mathfrak{h}=3,\text{ }\dim_{FL}\mathcal{Q}_{2}=9,\text{
}\dim_{FL}\mathcal{Q}_{3}=20,$ $\displaystyle\dim_{FL}\mathcal{Q}_{4}=$
$\displaystyle\text{ }30,\text{ }\dim_{FL}\mathcal{Q}_{5}=30,\text{
}\dim_{FL}\mathcal{Q}_{6}=12.$

Hence, $\dim_{FL}\mathcal{Q}_{B_{3}}(d)=104.$ We now write down the generic
formula for the non-trivial commutator relations. Without writing each
commutator explicitly, let us assume that the explicit generators above in
each of the six layer subsets is composed by

$\displaystyle\\{A_{1},A_{2},A_{3}\\}\cup\left\\{B_{1},\ldots,B_{9}\right\\}\cup\ldots\cup\left\\{E_{1},\ldots,E_{30}\right\\}\cup\left\\{F_{1},\ldots,F_{12}\right\\}:=\textbf{A}\cup\textbf{B}\cup\textbf{C}\cup\ldots\cup\textbf{E}\cup\textbf{F}.$

Here the element of degree one is indicated by the letter $A_{i},$ following
the alphabetical orders, elements of degree two with $B_{j}$ and so on. The
representatives for each subset have already been given in the construction
above. Then, up to the degree for which the polynomial algebra
$\mathcal{Q}_{B_{3}}(d)$ is closed, the non-trivial commutator relations
formally adopts the following structure

(74)
$\displaystyle\begin{matrix}\\{\textbf{B},\textbf{B}\\}\sim\textbf{C}+\textbf{A}\textbf{B}+\textbf{A}^{3}\\\
\\{\textbf{B},\textbf{C}\\}\sim\textbf{D}+\textbf{B}^{2}+\textbf{A}\\{\textbf{B},\textbf{B}\\}\\\
\\{\textbf{B},\textbf{D}\\}\sim\textbf{E}+\textbf{BC}+\textbf{A}\\{\textbf{B},\textbf{C}\\}\\\
\\{\textbf{B},\textbf{E}\\}\sim\textbf{BD}+\textbf{C}^{2}+\\{\textbf{B},\textbf{D}\\}\\\
\\{\textbf{C},\textbf{E}\\}\sim\textbf{BE}+\textbf{CD}+\textbf{B}^{2}\textbf{C}+\textbf{A}\\{\textbf{B},\textbf{E}\\}\\\
\\{\textbf{D},\textbf{E}\\}\sim\textbf{D}^{2}+\textbf{CE}+\textbf{B}\textbf{C}^{2}+\textbf{B}^{4}+\textbf{A}\\{\textbf{C},\textbf{E}\\}\\\
\\{\textbf{E},\textbf{E}\\}\sim\textbf{C}^{3}+\textbf{DE}+\textbf{B}^{3}\textbf{C}+\textbf{A}\\{\textbf{D},\textbf{E}\\}\\\
\\{\textbf{F},\textbf{E}\\}\sim\textbf{E}^{2}+\textbf{DF}+\textbf{B}^{5}+\textbf{A}\\{\textbf{E},\textbf{E}\\}\\\
\\{\textbf{F},\textbf{F}\\}\sim\textbf{EF}+\textbf{B}\textbf{C}^{3}+\textbf{C}\textbf{D}^{2}+\textbf{A}\\{\textbf{E},\textbf{F}\\}.\end{matrix}$

In particular, the multiplication of the terms are given by

$\displaystyle\textbf{BC}\sim\sum_{j=1}^{9}\sum_{k=1}^{20}c_{jk}^{pq}B_{j}C_{k}\text{
with }1\leq p,q\leq 20.$

Progressively expand the terms in $\eqref{eq:65}$, we obtain e.g.

$\displaystyle\\{\textbf{E},\textbf{E}\\}\sim\text{ }$
$\displaystyle\textbf{C}^{3}+\textbf{DE}+\textbf{B}^{3}\textbf{C}+\textbf{A}\left(\textbf{D}^{2}+\textbf{CE}+\textbf{B}\textbf{C}^{2}+\textbf{B}^{4}\right)+\textbf{A}^{2}\\{\textbf{C},\textbf{E}\\}$
$\displaystyle\sim\text{ }$
$\displaystyle\textbf{C}^{3}+\textbf{DE}+\textbf{B}^{3}\textbf{C}+\textbf{A}\left(\textbf{D}^{2}+\textbf{CE}+\textbf{B}\textbf{C}^{2}+\textbf{B}^{4}\right)+\textbf{A}^{2}(\textbf{BE}+\textbf{CD}+\textbf{B}^{2}\textbf{C})+\textbf{A}^{3}\\{\textbf{B},\textbf{E}\\}$
$\displaystyle\sim\text{ }$
$\displaystyle\textbf{C}^{3}+\textbf{DE}+\textbf{B}^{3}\textbf{C}+\textbf{A}\left(\textbf{D}^{2}+\textbf{CE}+\textbf{B}\textbf{C}^{2}+\textbf{B}^{4}\right)+\textbf{A}^{2}(\textbf{BE}+\textbf{CD}+\textbf{B}^{2}\textbf{C})+\textbf{A}^{3}(\textbf{BD}+\textbf{C}^{2})$
$\displaystyle+\textbf{A}^{4}\\{\textbf{B},\textbf{D}\\}$
$\displaystyle\sim\text{ }$
$\displaystyle\textbf{C}^{3}+\textbf{DE}+\textbf{B}^{3}\textbf{C}+\textbf{A}\left(\textbf{D}^{2}+\textbf{CE}+\textbf{B}\textbf{C}^{2}+\textbf{B}^{4}\right)+\textbf{A}^{2}(\textbf{BE}+\textbf{CD}+\textbf{B}^{2}\textbf{C})+\textbf{A}^{3}(\textbf{BD}+\textbf{C}^{2})$
$\displaystyle+\textbf{A}^{4}(\textbf{E}+\textbf{BC})+\textbf{A}^{5}(\textbf{D}+\textbf{B}^{2})+\textbf{A}^{6}\textbf{C}+\textbf{A}^{7}\textbf{B}+\textbf{A}^{9}$

Clearly, some of the coefficients listed above will be zero. In any case, it
is clear that $d\leq 5.$ Based on an explicit calculation, we obtain for
example

$\displaystyle\left\\{p_{II}^{\prime},\hat{p}_{II}^{\prime}\right\\}=$
$\displaystyle\text{
}\underbrace{\left\\{\epsilon_{l_{u}k}^{-},\epsilon_{kl_{u}}^{-}\right\\}_{o}p_{kl^{+},\hat{kl}^{+}}p_{lk_{u}^{+},\hat{lk_{u}}^{+}}p_{k_{u}l_{u}^{+},\hat{k_{u}l_{u}}^{+}}\left(p_{l,\hat{l}}\right)^{2}+\ldots+\left\\{\epsilon_{k_{u}l_{u}}^{+},\hat{\epsilon}_{k_{u}l_{u}}^{+}\right\\}_{o}p_{l_{u}k^{-},kl_{u}^{-}}p_{kl^{+},\hat{kl}^{+}}p_{lk_{u}^{+},\hat{lk_{u}}^{+}}\left(p_{l,\hat{l}}\right)^{2}}_{\text{$\in\textbf{A}\textbf{B}^{5}$}}$
$\displaystyle+\ldots$

This ensures that the degree of the polynomial algebra is $5.$ That is,
$\left(\mathcal{Q}_{B_{3}}(5),\\{\cdot,\cdot\\}\right)$ is a quintic algebra.

###  5.2. Explicit generators for $\mathcal{Q}_{D_{3}}(d^{\prime})$

Consider now polynomial (Poisson) algebras arising from the semisimple Lie
algebra of type $D_{3}$, i.e., the Cartan centralizer in $U(D_{3})$. The
Cartan subalgebra of $\mathfrak{so}(6,\mathbb{C})$ contains $3$ elements
$H_{1},H_{2},H_{3}.$ Take
$(\mathfrak{n}^{+})^{*}=\mathrm{span}\left\\{\epsilon_{12}^{-},\epsilon_{13}^{-},\epsilon_{23}^{-},\epsilon_{12}^{+},\epsilon_{13}^{+},\epsilon_{23}^{+}\right\\}$
and
$(\mathfrak{n}^{-})^{*}=\mathrm{span}\left\\{\epsilon_{21}^{-},\epsilon_{31}^{-},\epsilon_{32}^{-},\right.$
$\left.\hat{\epsilon}_{12}^{+},\hat{\epsilon}_{13}^{+},\hat{\epsilon}_{23}^{+}\right\\}$
such that
$\mathfrak{so}^{*}(6,\mathbb{C})=\mathrm{span}\\{h_{1},h_{2},h_{3}\\}\oplus(\mathfrak{n}^{-})^{*}\oplus(\mathfrak{n}^{+})^{*}.$
Applying the observation in Section 4.2, a short computation shows that there
are $23$ indecomposable solutions in the centralizer
$S\left(D_{3}\right)^{\mathfrak{h}}.$

We now proceed to construct these elements. From the construction, we know
that $\mathfrak{h}\subset\mathcal{Q}_{D_{3}}(d^{\prime}).$ Moreover, under
case (a), assume that $p\in\mathcal{Q}_{D_{3}}(d^{\prime})$ has the form of
either $g(\textbf{x}_{1})$ or $m_{1}(\textbf{x}_{L})$, where
$\textbf{x}_{1}=\left(\epsilon_{12}^{-},\epsilon_{13}^{-},\epsilon_{23}^{-},\epsilon_{21}^{-},\epsilon_{31}^{-},\epsilon_{32}^{-}\right)$
and
$\textbf{x}_{L}=\left(\epsilon_{12}^{+},\epsilon_{13}^{+},\epsilon_{23}^{+},\hat{\epsilon}_{12}^{+},\hat{\epsilon}_{13}^{+},\hat{\epsilon}_{23}^{+}\right)$.
Then $p\in\mathcal{Q}_{2}$ or $\mathcal{Q}_{3}.$ In particular, we have

$\displaystyle p_{12^{-},21^{-}}=\epsilon_{12}^{-}\epsilon_{21}^{-},\quad
p_{[13^{-}],31^{-}}=\left[\epsilon_{13}^{-}\right]\epsilon_{31}^{-},\quad
p_{[23^{-}],32^{-}}=\left[\epsilon_{23}^{-}\right]\epsilon_{32}^{-}\in\mathcal{Q}_{A_{2}}(2),$
(75) $\displaystyle
p_{12^{+},\hat{12}^{+}}=\epsilon_{12}^{+}\hat{\epsilon}_{12}^{+},\quad
p_{13^{+},\hat{13}^{+}}=\epsilon_{13}^{+}\hat{\epsilon}_{13}^{+},\quad
p_{23^{+},\hat{23}^{+}}=\epsilon_{23}^{+}\hat{\epsilon}_{23}^{+}.$

From the observation (b) in Subsection 4.2 and the discussion in Subsection
$\ref{5.1},$ taking into account the form of $m_{1}$ in $\eqref{eq:m1},$

(76)
$\displaystyle\begin{matrix}p_{[12^{-}];23^{+},\hat{13}^{+}}=\left[\epsilon_{12}^{-}\right]\epsilon_{23}^{+}\hat{\epsilon}_{13}^{+},\quad
p_{[13^{-}];23^{+},\hat{12}^{+}}=\left[\epsilon_{13}^{-}\right]\epsilon_{23}^{+}\hat{\epsilon}_{12}^{+},\quad
p_{[23^{-}];13^{+},\hat{12}^{+}}=\left[\epsilon_{23}^{-}\right]\epsilon_{13}^{+}\hat{\epsilon}_{12}^{+},\\\
p_{[21^{-}];13^{+},\hat{23}^{+}}=\left[\epsilon_{21}^{-}\right]\epsilon_{13}^{+}\hat{\epsilon}_{23}^{+},\quad
p_{[31^{-}];12^{+},\hat{23}^{+}}=\left[\epsilon_{31}^{-}\right]\epsilon_{12}^{+}\hat{\epsilon}_{23}^{+},\quad
p_{[32^{-}];12^{+},\hat{13}^{+}}=\left[\epsilon_{32}^{-}\right]\epsilon_{12}^{+}\hat{\epsilon}_{13}^{+},\end{matrix}$

are indecomposable monomials from the layers $\mathcal{Q}_{3}$ and
$\mathcal{Q}_{4}.$ Here the permutation root vectors in $\left[\cdot\right]$
are the same as in $\eqref{eq:id5520}$. These indecomposable integrals in
$\eqref{eq:so61}$ and $\eqref{eq:so62}$ generate
$\mathcal{Q}_{D_{3}}(d^{\prime})=\mathfrak{h}^{*}\oplus\mathcal{Q}_{2}\oplus\mathcal{Q}_{3}\oplus\mathcal{Q}_{4}$
with

$\displaystyle\dim\mathcal{Q}_{1}=\dim_{FL}\mathfrak{h}=3,\text{
}\dim_{FL}\mathcal{Q}_{2}=6,\text{ }\dim_{FL}\mathcal{Q}_{3}=8,\text{
}\dim_{FL}\mathcal{Q}_{4}=6,$

Hence, $\dim_{FL}\mathcal{Q}_{D_{3}}(d^{\prime})=23.$ Since $A_{3}\cong D_{3}$
are isomorphic as Lie algebras, it is natural to ask the relations between the
polynomial algebra $\mathcal{Q}_{D_{3}}(d^{\prime})$ and
$\mathcal{Q}_{A_{3}}(4).$ From [6], we observe that
$\dim_{FL}\mathcal{Q}_{D_{3}}(d^{\prime})=\dim_{FL}\mathcal{Q}_{A_{3}}(4)=23.$
The maximal number of functionally independent integrals is
$\mathcal{N}(\mathfrak{g})=\dim_{FL}\mathfrak{g}-3=12.$ In particular, the
functionally dependent relations from the generators above are

$\displaystyle p_{ij^{-},ji^{-}}p_{ik^{-},ki^{-}}p_{kj^{-},jk^{-}}=$
$\displaystyle\text{ }p_{[ij]^{-},ji^{-}}p_{[ik^{-}],ki^{-}}$ $\displaystyle
p_{ij^{-};jk^{+},\hat{ik}^{+}}p_{jk^{-};ik^{+},\hat{ij}^{+}}=$
$\displaystyle\text{ }p_{ik^{-};ij^{+},\hat{jk}^{+}}p_{ik^{+},\hat{ik}^{+}}$
$\displaystyle
p_{ik^{-},kj^{-};jk^{+},\hat{ik}^{+}}p_{ji^{-},ik^{-};ik^{+},\hat{ij}^{+}}=$
$\displaystyle\text{
}p_{ik^{-};jk^{+},\hat{ij}^{+}}p_{ik^{+},\hat{ik}^{+}}p_{ij^{-},ji^{-}}$
$\displaystyle p_{jk^{-};ik^{+},\hat{ij}^{+}}p_{kj^{-};ij^{+},\hat{ik}^{+}}=$
$\displaystyle\text{
}p_{kj^{-},jk^{-}}p_{ij^{+},\hat{ij}^{+}}p_{ik^{+},\hat{ik}^{+}}$

with all $1\leq i,k,j\leq 3.$ For instance,

$\displaystyle p_{12^{-};23^{+},\hat{13}^{+}}p_{23^{-};13^{+},\hat{12}^{+}}=$
$\displaystyle\text{ }p_{13^{-};23^{+},\hat{12}^{+}}p_{13^{+},\hat{13}^{+}},$
$\displaystyle p_{12^{-};23^{+},\hat{13}^{+}}p_{31^{-};12^{+},\hat{23}^{+}}=$
$\displaystyle\text{ }p_{32^{-};12^{+},\hat{13}^{+}}p_{23^{+},\hat{23}^{+}},$
$\displaystyle p_{13^{-};23^{+},\hat{12}^{+}}p_{21^{-};13^{+},\hat{23}^{+}}=$
$\displaystyle\text{ }p_{23^{-};13^{+},\hat{12}^{+}}p_{23^{+},\hat{23}^{+}},$
$\displaystyle p_{23^{-};13^{+},\hat{12}^{+}}p_{31^{-};12^{+},\hat{23}^{+}}=$
$\displaystyle\text{ }p_{21^{-};13^{+},\hat{23}^{+}}p_{12^{+},\hat{12}^{+}}.$

As functionally independent elements we can take

$\displaystyle
S=\left\\{h_{1},h_{2},h_{3},p_{12^{-},21^{-}},p_{13^{-},31^{-}},p_{23^{-},32^{-}},p_{12^{-},23^{-},31^{-}},p_{12^{+},\hat{12}^{+}},p_{13^{+},\hat{13}^{+}},p_{23^{+},\hat{23}^{+}},p_{12^{-};23^{+},\hat{13}^{+}},p_{23^{-};13^{+},\hat{12}^{+}}\right\\}.$

Under a Hamiltonian
$\mathcal{H}=\sum_{i_{1}+i_{2}+i_{3}=2}\Gamma_{i_{1},i_{2},i_{3}}h_{1}^{i_{1}}h_{2}^{i_{2}}h_{3}^{i_{3}}$,
the set $S$ forms a superintegrable system, where $\Gamma_{i_{1},i_{2},i_{3}}$
are some constants.

In analogy to the construction in Subsection $\ref{5.1},$ we can regroup the
generators into the sets
$\textbf{A}\cup\textbf{B}\cup\textbf{C}\cup\textbf{D}$. The non-trivial
Poisson brackets in terms of compact notations are thus given by

(77)
$\displaystyle\begin{matrix}\\{\textbf{B},\textbf{B}\\}\sim\textbf{C}+\textbf{A}\textbf{B}+\textbf{A}^{3}\\\
\\{\textbf{B},\textbf{C}\\}\sim\textbf{D}+\textbf{B}^{2}+\textbf{A}\\{\textbf{B},\textbf{B}\\}\\\
\\{\textbf{B},\textbf{D}\\}\sim\textbf{E}+\textbf{BC}+\textbf{A}\\{\textbf{B},\textbf{C}\\}\\\
\\{\textbf{C},\textbf{D}\\}\sim\textbf{BD}+\textbf{C}^{2}+\textbf{B}^{3}+\\{\textbf{B},\textbf{D}\\}\\\
\\{\textbf{D},\textbf{D}\\}\sim\textbf{CD}+\textbf{B}^{2}\textbf{C}+\textbf{A}\\{\textbf{C},\textbf{D}\\}\\\
\end{matrix}$

The last expression in $\eqref{eq:ma67}$ can further be expanded up as follows

$\displaystyle\\{\textbf{D},\textbf{D}\\}\sim\textbf{CD}+\textbf{B}^{2}\textbf{C}+\textbf{A}(\textbf{C}^{2}+\textbf{BD}+\textbf{B}^{3})+\textbf{A}^{2}\textbf{BC}+\textbf{A}^{3}(\textbf{D}+\textbf{B}^{2})+\textbf{A}^{4}\textbf{C}+\textbf{A}^{5}\textbf{B}+\textbf{A}^{7}.$

Using the explicit generators given above, it can be shown that
$\mathcal{Q}_{D_{3}}(3)$ is a cubic algebra. As an example, the explicit
expressions of the bilinear operation from $\\{\textbf{B},\textbf{B}\\}$ and
$\\{\textbf{B},\textbf{C}\\}$ are given by

$\displaystyle\\{p_{12^{-},21^{-}},p_{[13^{-}],31^{-}}\\}$
$\displaystyle=\left(p_{12^{-},23^{-},31^{-}}-p_{12^{-};23^{+},\hat{31}^{+}}\right)=-\\{p_{12^{-},21^{-}},p_{23^{-},32^{-}}\\},$
$\displaystyle\\{p_{12^{-},21^{-}},p_{13^{+},\hat{13}^{+}}\\}$
$\displaystyle=\left(p_{12^{-},23^{-},31^{-}}-p_{12^{-};23^{+},\hat{31}^{+}}\right)=-\left\\{p_{12^{-},21^{-}},p_{13^{+},\hat{13}^{+}}\right\\}$
$\displaystyle\\{p_{12^{-},21^{-}},p_{12^{-},23^{-},31^{-}}\\}$
$\displaystyle=(h_{2}-h_{1})p_{12^{-},23^{-},31^{-}}+p_{12^{-},21^{-}}\left(p_{[13^{-}],31^{-}}-p_{23^{-},32^{-}}\right)$
$\displaystyle\\{p_{12^{-},21^{-}},p_{21^{-};13^{+},\hat{32}^{+}}\\}$
$\displaystyle=(h_{2}-h_{1})p_{21^{-};13^{+},\hat{32}^{+}}+p_{12^{-},21^{-}}\left(p_{13^{+},\hat{13}^{+}}-p_{23^{+},\hat{23}^{+}}\right)$
$\displaystyle\\{p_{12^{-},21^{-}},p_{12^{-};23^{+},\hat{31}^{+}}\\}$
$\displaystyle=\left(p_{23,12}-p_{13^{-},21^{-};21^{+},\hat{31}}\right)=\left\\{p_{12^{-},21^{-}},p_{23^{-};12^{+},\hat{13}^{+}}\right\\},$
$\displaystyle\\{p_{12^{-},21^{-}},p_{21,3}\\}$
$\displaystyle=(h_{1}-h_{2})p_{21,3}+p_{12^{-},21^{-}}\left(p_{23^{+},\hat{23}^{+}}-p_{13^{+},\hat{13}^{+}}\right)$
$\displaystyle\\{p_{12^{-},21^{-}},p_{13^{-};32^{+},\hat{21}}\\}$
$\displaystyle=-\left(p_{12^{-},13^{-};31^{+},\hat{12}^{+}}+p_{12,23}\right)=\left\\{p_{12^{-},21^{-}},p_{23^{-};12^{+},\hat{13}^{+}}\right\\}$
$\displaystyle\left\\{p_{12^{-},21^{-}},p_{23,12}\right\\}$
$\displaystyle=p_{13^{-},21^{-};21^{+},\hat{31}}(h_{2}-h_{1})+p_{12^{-},21^{-}}\left(\left(p_{13^{-};32^{+},\hat{21}}\right)^{2}+p_{23^{-};12^{+},\hat{13}^{+}}\right),$
$\displaystyle\\{p_{12^{-},21^{-}},p_{12^{-},13^{-};31^{+},\hat{12}^{+}}\\}$
$\displaystyle=(h_{2}-h_{1})p_{12^{-},13^{-};31^{+},\hat{12}^{+}}-p_{12^{-},21^{-}}\left(\left(p_{31,2}\right)^{2}+p_{32^{-};21^{+},\hat{31}^{+}}\right)$

It is worthy to be mentioned that, as a consequence of the isomorphism
$A_{3}\simeq D_{3}$, the polynomial algebras are, as expected, equivalent, in
particular, of the same dimension and order, although they are not identical
in the generators, due to the different choice of basis.

###  5.3. Explicit generators for $\mathcal{Q}_{C_{3}}(d^{\prime\prime})$

As is well known, the Coxeter diagram for the root systems of
$\mathfrak{sp}(6,\mathbb{C})$ and $\mathfrak{so}(7,\mathbb{C})$ is the same,
with the difference being that short and long roots are interchanged [23].
This allows us to omit the observations (a) and (b) from Section $\ref{4.3}.$
Suppose that the coordinate ring of $\mathfrak{sp}(6,\mathbb{C})$ is given by
$(h_{1},h_{2},h_{3},\epsilon_{12}^{\pm},\epsilon_{13}^{\pm},\epsilon_{23}^{\pm},\hat{\epsilon}_{12}^{\pm},\hat{\epsilon}_{13}^{\pm},\hat{\epsilon}_{23}^{\pm},\epsilon_{1},\epsilon_{2},\epsilon_{3},\hat{\epsilon}_{1},\hat{\epsilon}_{2},\hat{\epsilon}_{3})$.

Explicit formulas for the generators in each subspace of
$\mathcal{Q}_{C_{3}}(d^{\prime\prime})$ are obtained via the classification in
Subsection 4.3. We first examine the polynomials that contain the short root
vectors. Based on observation (c), and using the formula $\eqref{eq:id577}$,
the monomials span $\mathcal{SK}_{2}$ are

(78)
$\displaystyle\left.\begin{matrix}p_{[12^{-}]^{2};\hat{1},2}=\left(\left[\epsilon_{12}^{-}\right]\right)^{2}\hat{\epsilon}_{1}\epsilon_{2},\quad
p_{[21^{-}]^{2};1,\hat{2}}=\left(\left[\epsilon_{21}^{-}\right]\right)^{2}\epsilon_{1}\hat{\epsilon}_{2}\\\
p_{[13^{-}]^{2};\hat{1},3}=\left(\left[\epsilon_{13}^{-}\right]\right)^{2}\hat{\epsilon}_{1}\epsilon_{3},\quad
p_{[31^{-}]^{2};1,\hat{3}}=\left(\left[\epsilon_{31}^{-}\right]\right)^{2}\epsilon_{1}\hat{\epsilon}_{3}\\\
p_{[23^{-}]^{2};\hat{2},3}=\left(\left[\epsilon_{23}^{-}\right]\right)^{2}\hat{\epsilon}_{2}\epsilon_{3},\quad
p_{[32^{-}]^{2};2,\hat{3}}=\left(\left[\epsilon_{32}^{-}\right]\right)^{2}\epsilon_{2}\hat{\epsilon}_{3}\end{matrix}\right\\}\in\mathcal{Q}_{4}\text{
or }\mathcal{Q}_{6}.$

Here, the elements in the set $[\cdot]$ are the same as $\eqref{eq:id5520}$.

We now have a look at all the indecomposable monomials that span the subspace
$\mathcal{K}_{1}\mathcal{K}_{2}.$ The procedure to find indecomposable
monomials developed in Section 5.1 can also be applied here. It follows that
the monomial $m_{1}$ has the same formulation as given in
$\eqref{eq:classif}$, $\eqref{eq:m1}$ and $\eqref{eq:classif2}.$ We first
assume that $\alpha_{n}\notin R\left(m_{1}\right).$ From $\eqref{eq:m1}$, the
indecomposable monomials are

(79)
$\displaystyle\begin{matrix}p_{13^{+},\hat{23}^{+};\hat{1},2}=\left(m_{1}^{(c)}\right)^{2}\hat{\epsilon}_{1}\epsilon_{2},\quad
p_{13^{+},\hat{23}^{+};1,\hat{3}}=\left(\hat{m}_{1}^{(c)}\right)^{2}\epsilon_{1}\hat{\epsilon}_{2}\\\
p_{12^{+},\hat{23}^{+};\hat{1},3}=\left(m_{1}^{(a)}\right)^{2}\hat{\epsilon}_{1}\epsilon_{3},\quad
p_{23^{+},\hat{12}^{+};1,\hat{3}}=\left(\hat{m}_{1}^{(a)}\right)^{2}\epsilon_{1}\hat{\epsilon}_{3}\\\
p_{12^{+},\hat{13}^{+};\hat{2},3}=\left(m_{1}^{(b)}\right)^{2}\hat{\epsilon}_{2}\epsilon_{3},\quad
p_{13^{+},\hat{12}^{+};2,\hat{3}}=\left(\hat{m}_{1}^{(b)}\right)^{2}\epsilon_{2}\hat{\epsilon}_{3}\end{matrix}$

in $\mathcal{Q}_{6}.$ On the other hands, if $\alpha_{n}\in
R\left(m_{1}\right),$ then we can formulate $m_{1}$ from
$\eqref{eq:classif2}$, leading to the following indecomposable monomials

(80)
$\displaystyle\left.\begin{matrix}p_{12^{+};\hat{1},\hat{2}}=\left(\epsilon_{12}^{+}\right)^{2}\hat{\epsilon}_{1}\hat{\epsilon}_{2},\quad
p_{13^{+};\hat{1},\hat{3}}=\left(\epsilon_{13}^{+}\right)^{2}\hat{\epsilon}_{1}\hat{\epsilon}_{3},\quad
p_{23^{+};\hat{2},\hat{3}}=\left(\epsilon_{23}^{+}\right)^{2}\hat{\epsilon}_{2}\hat{\epsilon}_{3};\\\
p_{\hat{12}^{+};1,2}=\left(\hat{\epsilon}_{12}^{+}\right)^{2}\epsilon_{1}\epsilon_{2},\quad
p_{\hat{13}^{+};1,3}=\left(\hat{\epsilon}_{13}^{+}\right)^{2}\epsilon_{1}\epsilon_{3},\quad
p_{\hat{23}^{+};2,3}=\left(\hat{\epsilon}_{23}^{+}\right)^{2}\epsilon_{2}\epsilon_{3}\end{matrix}\right\\}\in\mathcal{Q}_{4}.$

Moreover, if $m_{1}(\textbf{x}_{L})$ has the form of (B2), then

(81)
$\displaystyle\begin{matrix}p_{13^{+},12^{+},\hat{23}^{+};\hat{1}}=\epsilon_{13}^{+}m_{1}^{(a)}\hat{\epsilon}_{1},\quad
p_{23^{+},12^{+},\hat{13}^{+};\hat{2}}=\epsilon_{23}^{+}m_{1}^{(b)}\hat{\epsilon}_{2},\quad
p_{12^{+},13^{+},\hat{23}^{+};\hat{1}}=\epsilon_{12}^{+}m_{1}^{(c)}\hat{\epsilon}_{1},\\\
p_{\hat{13}^{+},23^{+},\hat{12}^{+};1}=\hat{\epsilon}_{13}^{+}\hat{m}_{1}^{(a)}\epsilon_{1},\quad
p_{\hat{23}^{+},13^{+},\hat{12}^{+};2}=\hat{\epsilon}_{23}^{+}\hat{m}_{1}^{(b)}\epsilon_{2},\quad
p_{\hat{12}^{+},23^{+},\hat{13}^{+};1}=\hat{\epsilon}_{12}^{+}\hat{m}_{1}^{(c)}\epsilon_{1}\end{matrix}$

are indecomposable monomials in $\mathcal{Q}_{4}.$

Finally, using the observation (e) in subsection 4.2 and the argument from
subsection $\ref{5.1},$ we obtain that

$\displaystyle p_{I}=$ $\displaystyle\text{
}\left[g_{I}\right]\epsilon_{kl}^{+}\hat{\epsilon}_{lj}^{+}\epsilon_{a}\hat{\epsilon}_{b}\text{
with $k\neq j\in I_{3}$};$ (82) $\displaystyle p_{II}=$ $\displaystyle\text{
}\left[g_{II}\right]\epsilon_{kl}^{+}\hat{\epsilon}_{b}\text{ with $k\neq l\in
I_{3}$};$ $\displaystyle p_{II}^{\prime}=$ $\displaystyle\text{
}\left[g_{II}^{\prime}\right]\epsilon_{kl}^{+}\epsilon_{ij}^{+}\hat{\epsilon}_{jm}^{+}\hat{\epsilon}_{b}\text{
with $k,l,j,i\neq m,b\in I_{3}.$}$

We then provide the formula of generators under the classification above. More
precisely, if the homogeneous generators have the form of $p_{I},$ we have

(83)
$\displaystyle\begin{matrix}p_{32^{-},12^{-};12^{+},\hat{23}^{+};2,\hat{1}},\quad
p_{21^{-},23^{-};12^{+},\hat{23}^{+};3,\hat{2}},\quad
p_{21^{-},31^{-};12^{+},\hat{13}^{+};1,\hat{2}},\\\
p_{13^{-},12^{-};12^{+},\hat{13}^{+};3,\hat{1}},\quad
p_{31^{-},32^{-};13^{+},\hat{23}^{+};2,\hat{3}},\quad
p_{23^{-},13^{-};13^{+},\hat{23}^{+};3,\hat{1}}\end{matrix}$ (84)
$\displaystyle\begin{matrix}p_{23^{-},21^{-};\hat{12}^{+},23^{+};\hat{2},1},\quad
p_{12^{-},32^{-};\hat{12}^{+},23^{+};\hat{3},2},\quad
p_{13^{-},12^{-};\hat{12}^{+},13^{+};\hat{1},2},\\\
p_{31^{-},21^{-};\hat{12}^{+},13^{+};\hat{3},1},\quad
p_{23^{-},13^{-};\hat{13}^{+},23^{+};\hat{2},3},\quad
p_{31^{-},32^{-};\hat{13}^{+},23^{+};\hat{3},1}\end{matrix}$

are indecomposable monomials in $\mathcal{Q}_{6}$. Here
$p_{32^{-},12^{-};12^{+},\hat{23}^{+};2,\hat{1}}=\epsilon_{32}^{-}\epsilon_{12}^{-}m_{1}^{(a)}\epsilon_{2}\hat{\epsilon}_{1}.$
Now, consider that indecomposable polynomials have the form of $p_{II}$. For
any $1\leq k,l\leq 3,$

(85) $\displaystyle\left.\begin{matrix}p_{[12^{-}];12^{+};\hat{1}},\quad
p_{[21^{-}];12^{+};\hat{2}},\quad p_{13^{-},23^{-};12^{+};\hat{3}},\\\
p_{[13^{-}];13^{+};\hat{1}},\quad p_{21^{-},32^{-};13^{+};\hat{2}},\quad
p_{[31^{-}];13^{+};\hat{3}},\\\ p_{21^{-},31^{-};23^{+};\hat{1}},\quad
p_{[23^{-}];23^{+};\hat{2}}\quad
p_{[32^{-}];23^{+};\hat{3}}\end{matrix}\right\\}\in\mathcal{Q}_{3}\text{ or
}\mathcal{Q}_{4}$

with the $\hat{\cdot}$ actions on $\eqref{eq:71}$ are indecomposable
monomials. Finally, consider that the indecomposable homogeneous polynomial
has the form of $p_{II}^{\prime}$ defined in $\eqref{eq:form2}.$ Since
$\mathrm{rank}\,\mathfrak{g}=3,$ the formula for the long root vectors are

$\displaystyle\left(\epsilon_{ij}^{+}\right)^{2}\hat{\epsilon}_{jm}^{+}\text{
and }\epsilon_{im}^{+}\epsilon_{ij}^{+}\hat{\epsilon}_{jm}^{+}.$

It is clear that for any $g\in S_{n}$-span,
$g\epsilon_{im}^{+}\epsilon_{ij}^{+}\hat{\epsilon}_{jm}^{+}\hat{\epsilon}_{s_{1}}$
with $s_{1}=i\neq j\neq m\in I_{3}$ are decomposable. Hence, with the form of
$m_{1}=\left(\epsilon_{ij}^{+}\right)^{2}\hat{\epsilon}_{jm}^{+},$ the
indecomposable monomials are given by

(86)
$\displaystyle\begin{matrix}p_{32^{-};{12^{+}}^{2},\hat{23}^{+};\hat{1}},\quad
p_{31^{-};{12^{+}}^{2},\hat{13}^{+};\hat{2}},\quad
p_{23^{-};{13^{+}}^{2},\hat{23}^{+};\hat{1}}\end{matrix}\in\mathcal{Q}_{5}.$

Furthermore, the monomials in $\eqref{eq:id71}$ under the $\hat{\cdot}$ action
are still indecomposable.

In conclusion, the polynomial algebra in $U(C_{3})^{\mathfrak{h}}$ is
generated by the above monomials. In particular,
$\mathcal{Q}_{C_{3}}(d^{\prime\prime})=\mathfrak{h}\oplus\bigoplus_{j\geq
2}^{6}\mathcal{Q}_{j}$ with

$\displaystyle\dim_{FL}\mathcal{Q}_{1}=$ $\displaystyle\text{
}\dim\mathfrak{h}=3,\text{ }\dim_{FL}\mathcal{Q}_{2}=9,\text{
}\dim_{FL}\mathcal{Q}_{3}=26,$ $\displaystyle\dim_{FL}\mathcal{Q}_{4}=$
$\displaystyle\text{ }36,\text{ }\dim_{FL}\mathcal{Q}_{5}=6,\text{
}\dim_{FL}\mathcal{Q}_{6}=24.$

Hence $\dim_{FL}\mathcal{Q}_{C_{3}}(d^{\prime\prime})=104.$ Denoting the
generators with the compact notation
$\textbf{A}\cup\ldots\cup\textbf{E}\cup\textbf{F}$, the non-trivial Poisson
brackets of the highest degree is given by

(87)
$\displaystyle\\{\textbf{F},\textbf{F}\\}\sim\textbf{EF}+\textbf{B}\textbf{C}^{3}+\textbf{C}\textbf{D}^{2}+\textbf{A}\\{\textbf{E},\textbf{F}\\}.$

Consider a monomial with maximum degree. It can be shown that
$d^{\prime\prime}=5.$ For instance, take $p_{I}\in\mathcal{Q}_{6},$ where
$p_{I}$ is defined in $\eqref{eq:id79}$. By a direct calculation,

$\displaystyle\left\\{p_{I},\hat{p}_{I}\right\\}=\underbrace{[g_{I}]\left[\hat{g}_{I}\right]\left\\{\epsilon_{kl}^{+},\hat{\epsilon}_{kl}^{+}\right\\}_{o}p_{\hat{lj}^{+},lj^{+}}p_{a,\hat{a}}p_{b,\hat{b}}+\ldots+[g_{I}]\left[\hat{g}_{I}\right]p_{kl^{+},\hat{kl}^{+}}p_{\hat{lj}^{+},lj^{+}}p_{a,\hat{a}}\left\\{\epsilon_{b},\hat{\epsilon}_{b}\right\\}_{o}}_{\text{$\in\textbf{AB}^{5}$
}}+\ldots,$

implying that $d^{\prime\prime}=5.$

From the construction in Section 5.1, 5.2 and 5.3, we, in particular, deduce
some embeddings between the different polynomial algebras. Moreover,
$\dim_{FL}\mathcal{Q}_{B_{3}}(5)=\dim_{FL}\mathcal{Q}_{C_{3}}(5)=104$ and

(88)
$\mathcal{Q}_{A_{2}}(2)\subset\mathcal{Q}_{A_{3}}(3);\quad\mathcal{Q}_{A_{2}}(2)\subset\mathcal{Q}_{D_{3}}(3)\subset\mathcal{Q}_{B_{3}}(5);\quad\mathcal{Q}_{A_{2}}(2)\subset\mathcal{Q}_{C_{3}}(5).$

##  6\. Conclusions

In this work, the procedure proposed in [6] to determine the centralizer of
the Cartan subalgebra in the universal enveloping algebra of a complex semi-
simple Lie algebra $\mathfrak{g}$ has been extended to the remaining classical
series $B_{n}$, $C_{n}$ and $D_{n}$. The method is based on a detailed
analysis of the root system, taking into account that for these series, there
are two root lengths, for which reason the ansatz must be generalized to
properly recover the indecomposable polynomials that generate the commutant. A
generic algorithm to determine the number of (indecomposable) linearly
independent polynomials of a given degree has been given, as well as the
general outline of the classification of the polynomial algebras associated to
each root system, that uses properties of the Weyl group. The detailed proof
of the technical details will be given elsewhere [5]. As an illustration of
the method, the rank three Lie algebras $B_{3}$, $C_{3}$ and $D_{3}$ have been
inspected in detail, also showing that, for the latter case, the isomorphism
with $A_{3}$ implies that the corresponding polynomial algebras are
equivalent, albeit their systems of generators are different, basing on the
choice of different bases. In particular, from the explicit presentation of
the polynomial algebras certain embeddings (88) can be deduced. In this
context, it should be recalled that the polynomial algebras related to the
root system of $A_{n}$ can be realized as higher-rank Racah algebras with a
suitable realization [6]. Therefore, the various embeddings of the polynomial
algebras $\mathcal{Q}_{A_{n}}(n)$,
$\mathcal{Q}_{B_{n}}(2n-1),\mathcal{Q}_{C_{n}}(2n-1)$ and
$\mathcal{Q}_{D_{n}}(2n-2)$, suggest that appropriate deformations of higher-
rank Racah algebras can provide an alternative interpretation of these
commutants, with potential applications in superintegrable systems on
manifolds more general than the spheres. On the other hand, as higher-rank
Racah algebras play an important role in recoupling theories, orthogonal
polynomials and superintegrability, also connected to models in the context of
Dunkl operators and Dirac equations, it is worthy to be analyzed in detail
whether the polynomial algebras related to $B_{n}$, $C_{n}$ and $D_{n}$ lead
to some physically relevant generalization. In order to complete the
description, the polynomial algebras related to root systems of exceptional
Lie algebras have still to be determined, with partial results for $G_{2}$
having already been obtained [10]. The problem for these cases is primarily of
computational nature, especially for the $E_{i}$ series, although it is
expected that the ansatz used for the classical series remains valid for the
exceptional root systems. It is hoped that the problem will soon be solved in
satisfactory manner. Work in this direction is currently in progress.

Summarizing, besides the intrinsic interest of polynomial algebras in
enveloping algebras and the construction of superintegrable systems for
adequate realizations of the generators by differential operators, the method
constitutes a unified approach to reinterpret symmetry algebras in a purely
algebraic setting.

## Acknowledgement

IM was supported by the Australian Research Council Future Fellowship
FT180100099. YZZ was supported by the Australian Research Council Discovery
Project DP190101529. RCS acknowledges financial support by the research grant
PID2019- 106802GB-I00/AEI/10.13039/501100011033 (AEI/ FEDER, UE).

## References

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