# Darboux families and the classification of real four-dimensional
indecomposable coboundary Lie bialgebras

J. de Lucas and D. Wysocki

###### Abstract

This work introduces a new concept, the so-called Darboux family, which is
employed to determine, to analyse geometrically, and to classify up to Lie
algebra automorphisms, in a relatively easy manner, coboundary Lie bialgebras
on real four-dimensional indecomposable Lie algebras. The Darboux family
notion can be considered as a generalisation of the Darboux polynomial for a
vector field. The classification of $r$-matrices and solutions to classical
Yang-Baxter equations for real four-dimensional indecomposable Lie algebras is
also given in detail. Our methods can further be applied to general, even
higher-dimensional, Lie algebras. As a byproduct, a method to obtain matrix
representations of certain Lie algebras with a non-trivial center is
developed.

##  1 Introduction

Lie bialgebras [15, 17, 18, 29] appeared as a tool to study integrable systems
[20, 21]. A Lie bialgebra is a Lie algebra $\mathfrak{g}$ along with a Lie
bracket on its dual space $\mathfrak{g}^{*}$ that amounts to a cocycle in a
Chevalley-Eilenberg cohomology of $\mathfrak{g}$. Lie bialgebras have also
applications to quantum gravity [4, 8, 38, 39] and other research fields [15,
29, 43]. In differential geometry, they occur in the problem of classifying
Poisson Lie groups [3, 15].

Although much research has been devoted to the classification of Lie
bialgebras up to Lie algebra automorphisms, there are still many open
problems. More specifically, Lie bialgebras on two- and three-dimensional Lie
algebras $\mathfrak{g}$ have been completely classified [22, 25]. Particular
instances of Lie bialgebras with $\dim\mathfrak{g}>3$, e.g. for a semi-simple
$\mathfrak{g}$, have also been studied [2, 11, 14, 30, 36, 43, 44]. Employed
techniques are rarely non-algebraic (cf. [35]) and they are not very effective
to analyse Lie bialgebras when $\dim\mathfrak{g}>3$ [2, 15, 22, 25, 35]. This
motivates the search for new approaches to the study and classification of Lie
bialgebras [35]. In particular, we are here interested in geometric
approaches.

A coboundary Lie bialgebra is a particular type of Lie bialgebra that is
characterised by means of an $r$-matrix, namely a bivector on a Lie algebra
$\mathfrak{g}$ that is a solution to its modified classical Yang-Baxter
equation (mCYBE) [15, 25]. This work introduces novel geometric techniques to
classify, up to Lie algebra automorphisms, coboundary Lie bialgebras and their
$r$-matrices on a fixed Lie algebra $\mathfrak{g}$.

In particular, we introduce a generalisation, the so-called Darboux family, of
the notion of Darboux polynomial for a polynomial vector field. On a finite-
dimensional vector space $E$, a polynomial function is a polynomial expression
on a set of linear coordinates on $E$. A Darboux polynomial, $P$, for a
polynomial vector field $X$ on $E$ is a polynomial function on $E$ so that
$XP=fP$ for a certain polynomial $f$ on $E$, the so-called cofactor of $P$
relative to $X$ [16, 33]. It is worth noting that, geometrically, one cannot
define intrinsically what a polynomial on a general manifold is, as the
explicit form of a function on a manifold depends on the chosen coordinate
system.

More generally, let us consider a $q$-dimensional Lie algebra of vector
fields, let us say $V:=\langle X_{1},\ldots,X_{q}\rangle$, on a manifold $M$.
A Darboux family for $V$ is an $s$-dimensional vector space
$\mathcal{A}:=\langle f_{1},\ldots,f_{s}\rangle$ of smooth functions on $M$
such that
$X_{\alpha}f_{\beta}=\sum_{\gamma=1}^{s}g_{\alpha\beta\gamma}f_{\gamma}$ for
certain smooth functions $g_{\alpha\beta\gamma}\in C^{\infty}(M)$ with
$\alpha=1,\ldots,q$ and $\beta,\gamma=1,\ldots,s$. In this work, Darboux
families are studied and employed to classify up to Lie algebra automorphisms
the $r$-matrices and coboundary Lie bialgebras on real four-dimensional Lie
algebras that are indecomposable [50], i.e. they cannot be written as a direct
sum of two proper ideals, namely ideals that are different of the zero and the
total Lie algebra. Nevertheless, our methods can also be applied to any other
Lie algebra.

Let ${\rm Aut}(\mathfrak{g})$ stand for the Lie group of Lie algebra
automorphisms of a Lie algebra $\mathfrak{g}$. Then, ${\rm Aut}(\mathfrak{g})$
naturally acts on the space $\Lambda^{2}\mathfrak{g}$ of bivectors [1] of
$\mathfrak{g}$ and, more specifically, on the space
$\mathcal{Y}_{\mathfrak{g}}\subset\Lambda^{2}\mathfrak{g}$ of solutions to the
mCYBE on $\mathfrak{g}$ (see [15, 35]). The classes of equivalent $r$-matrices
(up to Lie algebra automorphisms of $\mathfrak{g}$) are given by the orbits of
the action of ${\rm Aut}(\mathfrak{g})$ on $\mathcal{Y}_{\mathfrak{g}}$.
Characterising and analysing such orbits is in general complicated even when
the form of ${\rm Aut}(\mathfrak{g})$ is explicitly known (see [22] for a
typical algebraic approach to Lie bialgebras on three-dimensional Lie
algebras). Most techniques in the literature are algebraic [22, 25]. Instead,
we use here a more geometrical approach. Let us sketch our main ideas. We use
a simple method (see Proposition 2.2 and Remark 2.1) to determine the Lie
algebra of fundamental vector fields, $V_{\mathfrak{g}}$, of the action of
${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$. Our technique does not
require to know the explicit form of ${\rm Aut}(\mathfrak{g})$. Instead, it is
enough to know the space of derivations on $\mathfrak{g}$, which can be
derived by solving a linear system of algebraic equations. We show that the
orbits of the connected part of ${\rm Aut}(\mathfrak{g})$ containing its
neutral element, ${\rm Aut}_{c}(\mathfrak{g})$, are the integral connected
submanifolds of the generalised distribution, $\mathscr{E}_{\mathfrak{g}}$,
spanned by the vector fields of $V_{\mathfrak{g}}$ (see [27, 31, 41, 48, 49,
51] for general results on generalised distributions). In fact, as
$V_{\mathfrak{g}}$ is a finite-dimensional Lie algebra,
$\mathscr{E}_{\mathfrak{g}}$ is integrable [31]. Finding the integral
connected submanifolds of a generalised distribution, its so-called strata, is
more complicated than finding the leaves of standard distributions because
strata cannot always be determined by a family of common first-integrals for
the elements of $V_{\mathfrak{g}}$, as it happens in the case of distributions
(cf. [1, 28]). We determine them here via our Darboux families, which is
relatively easy as illustrated by our examples. Once the strata of
$\mathscr{E}_{\mathfrak{g}}$ in $\mathcal{Y}_{\mathfrak{g}}$ have been
obtained, the use of the action on such strata of an element of each connected
part of ${\rm Aut}(\mathfrak{g})$ allows us to determine the orbits of ${\rm
Aut}(\mathfrak{g})$ on $\mathcal{Y}_{\mathfrak{g}}$, which gives us the
desired classification of $r$-matrices. Note that our method do rely on the
determination of a single element of each connected component of ${\rm
Aut}(\mathfrak{g})$ and, more relevantly, it does not need the explicit action
of the whole ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$.

It is known that two $r$-matrices for $\mathfrak{g}$ that are not equivalent
up to a Lie algebra automorphism may give rise to coboundary Lie bialgebras
that are equivalent up to a Lie algebra automorphism (cf. [15, 35]). Let
$(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}$ be the space of bivectors of
$\mathfrak{g}$ that are invariant relative to the action of elements of
$\mathfrak{g}$ via the algebraic Schouten bracket [15, 35]. In our work, we
prove that the orbits of ${\rm Aut}(\mathfrak{g})$ on
$\mathcal{Y}_{\mathfrak{g}}$ that project onto the same orbit of the natural
action of ${\rm Aut}(\mathfrak{g})$ on
$\Lambda^{2}\mathfrak{g}/(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}$ (see [35])
are exactly the $r$-matrices that lead to equivalent (up to Lie algebra
automorphisms) coboundary Lie bialgebras on $\mathfrak{g}$.

Our methods are computationally affordable for the study and classification of
general coboundary Lie bialgebras with three- and four-dimensional
$\mathfrak{g}$. Our techniques are quite probably appropriate for looking into
and classifying Lie bialgebras on any five-dimensional $\mathfrak{g}$ and,
possibly, for other particular instances of higher-dimensional Lie algebras
$\mathfrak{g}$. Indeed, our procedures lead to the classification of real
coboundary Lie bialgebras, up to Lie algebra automorphisms, on any four-
dimensional indecomposable Lie algebra $\mathfrak{g}$ (see [50] for a
classification of indecomposable Lie algebras up to dimension six). Our
results are summarised in Table 2. This is a remarkable advance relative to
other classification works in the literature [2, 22, 25]. Indeed, the most
complicated classification in the literature so far is probably the
classification of Lie bialgebras on symplectic four-dimensional Lie algebras
[2]. Additionally, other four-dimensional Lie bialgebras have been partially
studied in the mathematics and physics literature (see [4, 6, 7, 10, 12, 11,
30, 43, 44] and references therein). As a byproduct of our research, a method
for the matrix representation of a class of finite-dimensional Lie algebras
with a non-trivial center, which cannot be represented through the matrices of
the adjoint representation, is given. This is interesting not only for our
purposes, but also in many other works where such a representation is employed
in practical calculations, e.g. [5].

The structure of the paper goes as follows. Section 2 surveys the main notions
on Lie bialgebras and their derivations, $\mathfrak{g}$-modules, and the
notation to be used. In Section 3, the theory of generalised distributions is
discussed. A method to obtain a matrix representation of a class of Lie
algebras with non-trivial center is given in Section 4. Section 5 introduces
Darboux families and shows how they can be used to classify $r$-matrices on
the Lie algebra $\mathfrak{s}_{4,1}$ (according to Šnobl and Winternitz’s
notation in [50]), up to Lie algebra automorphisms thereof. Section 6 analyses
several geometric properties of solutions to mCYBEs and it also provides some
hints on the use Darboux families to study the equivalence up to Lie algebra
automorphisms of $r$-matrices and coboundary Lie bialgebras. In Section 7,
Darboux families are applied to studying and classifying coboundary Lie
bialgebras on real four-dimensional indecomposable Lie algebras. Section 8
summarises our results and presents some further work in progress.

##  2 Fundamentals on Lie bialgebras and their derivations

Let us provide a brief account on the notions of Lie bialgebras, Schouten
brackets, and $\mathfrak{g}$-modules to be used hereafter (see [15, 29, 51]
for further details). Some simple results on the geometric properties of the
action of ${\rm Aut}(\mathfrak{g})$ on spaces of $k$-vectors are provided. Our
approach is more geometric than in standard works on Lie bialgebras. We
hereafter assume that $\mathfrak{g}$ and $E$ are a finite-dimensional real Lie
algebra and a finite-dimensional real vector space, respectively. Meanwhile,
$GL(E)$ and $\mathfrak{gl}(E)$ stand for the Lie group of automorphisms and
the Lie algebra of endomorphisms on $E$, respectively.

Let $\mathcal{V}^{m}M$ be the vector space of $m$-vector fields on a manifold
$M$. The Schouten-Nijenhuis bracket [37, 51] on
$\mathcal{V}M:=\oplus_{m\in\mathbb{Z}}\mathcal{V}^{m}M$ is the unique bilinear
map $[\cdot,\cdot]:\mathcal{V}M\times\mathcal{V}M\to\mathcal{V}M$ satisfying
that

$[f,g]=0,$ $[X_{1}\wedge\ldots\wedge
X_{s},f]:=\iota_{df}X_{1}\wedge\ldots\wedge
X_{s}:=\sum_{i=1}^{s}(-1)^{i+1}(X_{i}f)X_{1}\wedge\ldots\wedge\widehat{X_{i}}\wedge\ldots\wedge
X_{s},$ $[X_{1}\wedge\ldots\wedge X_{s},Y_{1}\wedge\ldots\wedge
Y_{l}]\\!\\!:=\\!\\!\\!\\!\sum_{\begin{subarray}{c}i=1,\ldots,s\\\
j=1,\ldots,l\end{subarray}}\\!\\!\\!(-1)^{i+j}[X_{i},Y_{j}]\wedge
X_{1}\wedge\ldots\wedge\widehat{X}_{i}\wedge\ldots\wedge X_{s}\wedge
Y_{1}\wedge\ldots\wedge\widehat{Y}_{j}\wedge\ldots\wedge Y_{l},$ (2.1)

where $f,g\in C^{\infty}(M)$, $s,l\in\mathbb{N}$, the
$X_{1},\ldots,X_{s},Y_{1},\ldots,Y_{l}$ are arbitrary vector fields on $M$, an
omitted vector field $X$ is denoted by the hat symbol $\widehat{X}$, and
$[X_{i},Y_{j}]$ is the Lie bracket111We denote similar structures with the
same symbol, but their meaning is clear from the context. of $X_{i}$ and
$Y_{j}$ (see [37] for more details). Remarkably,
$[\mathcal{X},\mathcal{Y}]\in\mathcal{V}^{s+l-1}$ for
$\mathcal{X}\in\mathcal{V}^{s}M$ and $\mathcal{Y}\in\mathcal{V}^{l}M$. The
space, $\mathcal{V}^{L}G$, of left-invariant elements of $\mathcal{V}G$ for a
Lie group $G$ is closed relative to $[\cdot,\cdot]$. In particular, left-
invariant vector fields on $G$, i.e. ${\mathcal{V}^{L}}^{1}G$, span a finite-
dimensional Lie algebra called the Lie algebra of $G$, which can be identified
with $T_{e}G$ [1]. Vice versa, every abstract finite-dimensional Lie algebra
$\mathfrak{g}$ can be thought of as the Lie algebra of left-invariant vector
fields of a Lie group [19]. Meanwhile, $\mathcal{V}^{L}G$ can be identified
with the Grassmann algebra $\Lambda\mathfrak{g}$, namely the algebra relative
to the exterior product spanned by all the multivectors of the Lie algebra,
$\mathfrak{g}$, of $G$ [26]. Moreover, $[\cdot,\cdot]$ can be restricted to
$\mathcal{V}^{L}G$ leading to the algebraic Schouten bracket on
$\Lambda\mathfrak{g}$ [35, 51]. For simplicity, we will call it the Schouten
bracket on $\Lambda\mathfrak{g}$. Recall that the Grassmann algebra $\Lambda
E$ of a vector space $E$ satisfies that $\Lambda
E=\bigoplus_{m\in\mathbb{Z}}\Lambda^{m}E$, where $\Lambda^{m}E$ is the vector
space of $m$-vectors of $E$.

A Lie bialgebra is a pair $(\mathfrak{g},\delta)$, where $\mathfrak{g}$ admits
a Lie bracket $[\cdot,\cdot]_{\mathfrak{g}}$, whilst
$\delta:\mathfrak{g}\to\Lambda^{2}\mathfrak{g}$, the cocommutator, is a linear
map, its transpose $\delta^{*}:\Lambda^{2}\mathfrak{g}^{*}\to\mathfrak{g}^{*}$
is a Lie bracket on $\mathfrak{g}^{*}$, and

$\delta([v_{1},v_{2}]_{\mathfrak{g}})=[v_{1},\delta(v_{2})]+[\delta(v_{1}),v_{2}],\qquad\forall
v_{1},v_{2}\in\mathfrak{g}.$ (2.2)

A Lie bialgebra homomorphism is a Lie algebra homomorphism
$\phi:\mathfrak{g}\rightarrow\mathfrak{h}$ between Lie bialgebras
$(\mathfrak{g},\delta_{\mathfrak{g}})$ and
$(\mathfrak{h},\delta_{\mathfrak{h}})$ such that
$(\phi\otimes\phi)\circ\delta_{\mathfrak{g}}=\delta_{\mathfrak{h}}\circ\phi$.
A coboundary Lie bialgebra is a Lie bialgebra $(\mathfrak{g},\delta_{r})$ such
that $\delta_{r}(v):=[v,r]$ for every $v\in\mathfrak{g}$ and some
$r\in\Lambda^{2}\mathfrak{g}$, a so-called $r$-matrix. To characterise
$r$-matrices, we use the following notions and Theorem 2.1. The standard
identification of an abstract Lie algebra $\mathfrak{g}$ with the Lie algebra
of left-invariant vector fields on a Lie group $G$ allows us to understand the
tensor algebra $\mathfrak{T}(\mathfrak{g})$ of $\mathfrak{g}$ as the tensor
algebra, $\mathfrak{T}^{L}(G)$, of left-invariant tensor fields on $G$. This
gives rise to a Lie algebra representation
$\textrm{ad}:v\in\mathfrak{g}\mapsto{\rm
ad}_{v}\in\mathfrak{gl}\left(\mathfrak{T}(\mathfrak{g})\right)$, where ${\rm
ad}_{v}(w):=\mathcal{L}_{v}w$ for every $w\in\mathfrak{T}(\mathfrak{g})$ and
$\mathcal{L}_{v}w$ is the Lie derivative of $w$ relative to $v$, which are
understood as elements in $\mathfrak{T}^{L}(G)$ in the natural way. Note that
the geometric notation $\mathcal{L}_{v}w$ is conciser than algebraic ones (cf.
[15]). An element $q\in\mathfrak{T}(\mathfrak{g})$ is called
$\mathfrak{g}$-invariant if $\mathcal{L}_{v}q=0$ for all $v\in\mathfrak{g}$.
We write $(\mathfrak{T}(\mathfrak{g}))^{\mathfrak{g}}$ for the set of
$\mathfrak{g}$-invariant elements of $\mathfrak{T}(\mathfrak{g})$. The map
${\rm ad}:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{T}(\mathfrak{g}))$
admits a restriction ${\rm
ad}:\mathfrak{g}\rightarrow\mathfrak{gl}(\Lambda^{m}\mathfrak{g})$. We recall
that $(\Lambda^{m}\mathfrak{g})^{\mathfrak{g}}$ stands for the space of
$\mathfrak{g}$-invariant $m$-vectors. Let us recall the following well-known
result.

######  Theorem 2.1.

The map $\delta_{r}:v\in\mathfrak{g}\mapsto[v,r]\in\Lambda^{2}\mathfrak{g}$,
for $r\in\Lambda^{2}\mathfrak{g}$, is a cocommutator if and only if
$[r,r]\in(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$.

We call $[r,r]\in(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$ the modified
classical Yang-Baxter equation (mCYBE) of $\mathfrak{g}$, while $[r,r]=0$ is
referred to as the classical Yang-Baxter equation (CYBE) of $\mathfrak{g}$ and
its solutions amount to left-invariant Poisson bivectors on any Lie group $G$
with Lie algebra isomorphic to $\mathfrak{g}$ [51].

Note that two $r$-matrices $r_{1},r_{2}\in\Lambda^{2}\mathfrak{g}$ satisfy
that $\delta_{r_{1}}=\delta_{r_{2}}$ if and only if
$r_{1}-r_{2}\in(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}$. Then, what really
matters to the determination of coboundary Lie bialgebras is not $r$-matrices,
but their equivalence classes in the quotient space
$\Lambda^{2}_{R}\mathfrak{g}:=\Lambda^{2}\mathfrak{g}/(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}$.

A $\mathfrak{g}$-module is a pair $(V,\rho)$, where
$\rho:v\in\mathfrak{g}\mapsto\rho_{v}\in\mathfrak{gl}(V)$ is a Lie algebra
morphism. A $\mathfrak{g}$-module $(V,\rho)$ will be represented just by $V$,
while $\rho_{v}(x)$, for any $v\in\mathfrak{g}$ and $x\in V$, will be written
simply as $vx$ if $\rho$ is understood from the context. If ${\rm
ad}:v\in\mathfrak{g}\mapsto[v,\cdot]_{\mathfrak{g}}\in\mathfrak{gl}(\mathfrak{g})$
stands for the adjoint representation of $\mathfrak{g}$, then the fact that
each $[v,\cdot]_{\mathfrak{g}}$, with $v\in\mathfrak{g}$, is a derivation of
the Lie algebra $\mathfrak{g}$ (relative to its Lie bracket
$[\cdot,\cdot]_{\mathfrak{g}}$) allows us to ensure that $(\mathfrak{g},{\rm
ad})$ is a $\mathfrak{g}$-module [24]. The map ${\rm ad}$ can be considered as
a mapping of the form ${\rm
ad}:\mathfrak{g}\rightarrow\mathfrak{der}(\mathfrak{g})$, where
$\mathfrak{der}(\mathfrak{g})$ is the Lie algebra of derivations on
$\mathfrak{g}$. As a second relevant example of $\mathfrak{g}$-module,
consider the Lie group ${\rm Aut}(\mathfrak{g})$ of the Lie algebra
automorphisms of $\mathfrak{g}$ (see [45] for details on its Lie group
structure) and its Lie algebra, which is denoted by
$\mathfrak{aut}(\mathfrak{g})$. The tangent map at the identity map on
$\mathfrak{g}$, let us say ${\rm id}_{\mathfrak{g}}\in{\rm
Aut}(\mathfrak{g})$, to the injection $\iota:{\rm
Aut}(\mathfrak{g})\hookrightarrow GL(\mathfrak{g})$ induces a Lie algebra
morphism $\widehat{\rm ad}:\mathfrak{aut}(\mathfrak{g})\simeq{\rm T}_{{\rm
id}_{\mathfrak{g}}}{\rm
Aut}(\mathfrak{g})\rightarrow\mathfrak{gl}(\mathfrak{g})\simeq{\rm T}_{{\rm
id}_{\mathfrak{g}}}GL(\mathfrak{g})$ and $(\mathfrak{g},\widehat{\rm ad})$
becomes an $\mathfrak{aut}(\mathfrak{g}$)-module.

In view of the properties of the algebraic Schouten bracket, each
$\mathfrak{g}$ gives rise to a $\mathfrak{g}$-module
$(\Lambda\mathfrak{g},{\rm ad})$, where ${\rm
ad}:v\in\mathfrak{g}\mapsto[v,\cdot]\in\mathfrak{gl}(\Lambda\mathfrak{g})$
(cf. [51]). This fact can be viewed as a consequence of [35, Proposition 2.1].
To grasp this result and related ones, recall that every
$T\in\mathfrak{gl}(E)$ gives rise to the mappings
$\Lambda^{m}T:\lambda\in\Lambda^{m}E\mapsto 0\in\Lambda^{m}E$ for $m\leq 0$,
and the maps $\Lambda^{m}T\in\mathfrak{gl}(\Lambda^{m}E)$, for $m>0$, given by
the restriction to $\Lambda^{m}E$ of $\Lambda^{m}T:={{T\otimes{\rm
id}\otimes\ldots\otimes{\rm id}}}(m-{\rm operators})+\ldots+{{{\rm
id}\otimes\ldots\otimes{\rm id}\otimes T}}(m-{\rm operators}),$ where ${\rm
id}$ is the identity on $E$. Moreover, $\Lambda
T:=\bigoplus_{m\in\mathbb{Z}}\Lambda^{m}T\in\mathfrak{gl}(\Lambda E)$. If $T$
is considered as an element of $GL(E)$, we define
$\Lambda^{m}T:=T\otimes\ldots\otimes T(m-{\rm operators})$ for $m\geq 1$ and
$\Lambda^{m}T$ is the identity on $\Lambda^{m}E$ for $m\leq 0$. Finally,
$\Lambda T:=\bigoplus_{m\in\mathbb{Z}}\Lambda^{m}T$.

The Lie group ${\rm Aut}(\mathfrak{g})$ gives rise to a Lie group action
$T\in{\rm Aut}(\mathfrak{g})\mapsto\Lambda^{m}T\in
GL(\Lambda^{m}\mathfrak{g})$. Moreover, this gives rise to an infinitesimal
Lie group action
$\rho:\mathfrak{aut}(\mathfrak{g})\rightarrow\mathfrak{gl}(\Lambda^{m}\mathfrak{g})$
such that

$[\rho(d)](w):=\frac{d}{dt}\bigg{|}_{t=0}[\Lambda^{m}\exp(td)](w)=(\Lambda^{m}d)(w),\qquad\forall
d\in\mathfrak{aut}(\mathfrak{g}),\qquad\forall w\in\Lambda^{m}\mathfrak{g}.$

We write $V_{\mathfrak{g}}$ for the Lie algebra of fundamental vector fields
of this Lie group action for $m=2$. Recall that $\mathfrak{aut}(\mathfrak{g})$
is indeed the space of derivations on $\mathfrak{g}$ (cf. [35]). The following
result was proved in [35].

######  Proposition 2.1.

The dimension of the orbit $\mathcal{O}_{w}$ of the action of ${\rm
Inn}(\mathfrak{g})$ on $\Lambda^{m}\mathfrak{g}$ through
$w\in\Lambda^{m}\mathfrak{g}$ is $\dim{\rm Im}\,\Theta^{m}_{w},$ where
$\Theta^{m}_{w}:v\in\mathfrak{inn}(\mathfrak{g})\mapsto[v,w]\in\Lambda^{m}\mathfrak{g}$.

In this work, we will use the following rather straightforward generalisation
of Proposition 2.1.

######  Proposition 2.2.

The dimension of the orbit $\mathscr{O}_{w}$ of the action of ${\rm
Aut}(\mathfrak{g})$ on $\Lambda^{m}\mathfrak{g}$ through
$w\in\Lambda^{m}\mathfrak{g}$ is $\dim{\rm Im}\,\Upsilon^{m}_{w},$ where
$\Upsilon^{m}_{w}:d\in\mathfrak{der}(\mathfrak{g})\mapsto(\Lambda^{m}d)(w)\in\Lambda^{m}\mathfrak{g}$.

###### Proof.

The orbit of $w\in\Lambda^{m}\mathfrak{g}$ relative to ${\rm
Aut}(\mathfrak{g})$ is given by the points $(\Lambda^{m}T)(w)$ for every
$T\in{\rm Aut}(\mathfrak{g})$. Define $\exp(td)=:T_{t}$, with
$t\in\mathbb{R}$, for $d\in\mathfrak{der}(\mathfrak{g})$. Then, the tangent
space at $w$ of $\mathscr{O}_{w}$ is spanned by the tangent vectors

$\frac{d}{dt}\bigg{|}_{t=0}[\Lambda^{m}\exp(td)](w)=(\Lambda^{m}d)(w)\in
T_{w}\Lambda^{m}\mathfrak{g}\simeq\Lambda^{m}\mathfrak{g}.$ (2.3)

Then,
$\dim\mathscr{O}_{w}=\dim\mathfrak{der}(\mathfrak{g})-\dim\mathfrak{g}_{w}$,
where $G_{w}$ is the isotropy group of $w\in\Lambda^{m}\mathfrak{g}$ relative
to the action of ${\rm Aut}(\mathfrak{g})$ and $\mathfrak{g}_{w}$ is the Lie
algebra of $G_{w}$. Moreover, $\mathfrak{g}_{w}$ is given by those
$d\in\mathfrak{der}(\mathfrak{g})$ such that $(\Lambda^{m}d)(w)=0$. This
amounts to $d\in\ker\,\Upsilon^{m}_{w}$. Hence,
$\dim\mathscr{O}_{w}=\dim\mathfrak{der}(\mathfrak{g})-\dim\mathfrak{g}_{w}=\dim{\rm
Im}\,\Upsilon_{w}^{m}$. ∎

######  Remark 2.1.

As a byproduct, Proposition 2.2 shows that the fundamental vector fields of
the natural Lie group action of ${\rm Aut}(\mathfrak{g})$ on
$\Lambda^{2}\mathfrak{g}$ are spanned by
$X^{2}_{v}(w)=\sum_{i=1}^{s}[\Upsilon^{2}_{w}]_{i}\partial_{x_{i}}$, where
$x_{1},\ldots,x_{s}$ is any linear coordinate system on
$\Lambda^{2}\mathfrak{g}$, the $w$ is any point in $\Lambda^{2}\mathfrak{g}$,
and $v$ belongs to $\mathfrak{der}(\mathfrak{g})$. Moreover, the coordinates
of such vector fields are the coordinates of the extensions to
$\Lambda^{2}\mathfrak{g}$ of the derivations of $\mathfrak{g}$, which can
easily be obtained as the solutions, $d$, to the linear problem

$d([e_{1},e_{2}])=[d(e_{1}),e_{2}]+[e_{1},d(e_{2})],\qquad\forall
e_{1},e_{2}\in\mathfrak{g}.$

##  3 Generalised distributions

In this work we want to show that the problem of determining equivalent
$r$-matrices up to Lie algebra automorphisms can be significantly simplified
by studying the strata of the so-called generalised distributions [31, 48, 49,
51]. Let us detail some useful facts on these geometric entities. Unless
otherwise stated, we assume all objects to be smooth and globally defined.
Hereafter, $\mathfrak{X}(M)$ stands for the Lie algebra of vector fields on
$M$.

A generalised distribution (also called a Stefan–Sussmann distribution) on a
manifold $M$ is a correspondence $\mathcal{D}$ attaching each $x\in M$ to a
subspace $\mathcal{D}_{x}\subset T_{x}M$. We call rank of $\mathcal{D}$ at $x$
the dimension of $\mathcal{D}_{x}$. A generalised distribution need not have
the same rank at every point of $M$. If $\mathcal{D}$ has the same rank at
every point of $M$, then $\mathcal{D}$ is said to be regular or $\mathcal{D}$
is simply called a distribution. Otherwise, $\mathcal{D}$ is said to be
singular. A generalised distribution $\mathcal{D}$ on $M$ is _involutive_ if
every two vector fields taking values in $\mathcal{D}$ satisfy that their Lie
bracket takes values in $\mathcal{D}$ as well.

A _stratification_ , let us say $\mathcal{F}$, on a manifold $M$ is a
partition of $M$ into connected disjoint immersed submanifolds
$\\{\mathcal{F}_{k}\\}_{k\in I}$, where $I$ is a certain set of indices, i.e.
$M=\bigcup_{k\in I}\mathcal{F}_{k}$ and the submanifolds
$\\{\mathcal{F}_{k}\\}_{k\in I}$ satisfy
$\mathcal{F}_{k}\cap\mathcal{F}_{k^{\prime}}=\emptyset$ for $k\neq k^{\prime}$
and $k,k^{\prime}\in I$. The connected immersed submanifolds
$\mathcal{F}_{k}$, with $k\in I$, are called the strata of the stratification.
A stratification is _regular_ if its strata are immersed submanifolds of the
same dimension, whilst it is _singular_ otherwise. Regular stratifications are
called foliations and their strata are called leaves. The tangent space to a
stratum, $\mathcal{F}_{k}$, of a stratification passing through a point $x\in
M$ is a subspace $\mathcal{D}_{x}\subset T_{x}M$. All the subspaces
$\mathcal{D}_{x}\subset T_{x}M$ for every point $x\in M$ give rise to a
generalised distribution $\mathcal{D}:=\bigcup_{x\in M}\mathcal{D}_{x}$ on
$M$. All the leaves of a foliation have the same dimension and, therefore, the
generalised distribution formed by the tangent spaces at every point to its
leaves is regular. Meanwhile, a singular stratification gives rise to a
singular generalised distribution.

In this work, we are specially interested in generalised distributions
generated by finite-dimensional Lie algebras of vector fields, the so-called
Vessiot–Guldberg Lie algebras [34]. More specifically, let $V$ be a
Vessiot–Guldberg Lie algebra, the vector fields of $V$ span a generalised
distribution $\mathcal{D}^{V}$ given by

$\mathcal{D}_{x}^{V}:=\\{X_{x}:X\in V\\}\subset T_{x}M,\qquad\forall x\in M.$

Since the space of vector fields tangent to the strata of a stratification are
closed under Lie brackets, the Lie bracket of vector fields on $M$ taking
values in a distribution $\mathcal{D}$ can be restricted to each one of its
strata. A generalised distribution $\mathcal{D}$ on $M$ is _integrable_ if
there exists a stratification $\mathcal{F}$ on $M$ such that each stratum
$\mathcal{F}_{k}$ thereof satisfies $T_{x}\mathcal{F}_{k}=\mathcal{D}_{x}$ for
every $x\in\mathcal{F}_{k}$. A relevant question is whether a generalised
distribution on $M$ is integrable of not. For regular distributions, the
Frobenius theorem holds [23, 31].

######  Theorem 3.1.

If $\mathcal{D}$ is a distribution on a manifold $M$, then $\mathcal{D}$ is
integrable if and only if it is involutive.

Involutivity is a natural necessary condition for a generalised distribution
to be integrable because the set of vector fields tangent to any stratum of a
stratification is involutive. Nevertheless, if a generalised distribution is
not regular, its involutiveness does not necessarily implies its
integrability. If a generalised distribution $\mathcal{D}$ on $M$ is
analytical, i.e. for every $x\in M$ there exists a family of analytical vector
fields taking values in $\mathcal{D}$ and spanning $\mathcal{D}_{x^{\prime}}$
for every $x^{\prime}$ in an open neighbourhood of $x$, one has the following
proposition.

######  Theorem 3.2.

_(Nagano[41] and [31])_ Let $M$ be a real analytic manifold, and let $V$ be a
sub-Lie algebra of analytic vector fields on $M$. Then, the induced analytic
distribution $\mathcal{D}^{V}$ is integrable.

If a generalised distribution is not analytical, the Stefan-Sussmann’s theorem
establishes additional conditions to involutivity to ensure integrability. We
are more interested in the following result that ensures integrability in a
particular case of relevance to us.

######  Theorem 3.3.

(Hermann [27] and [31]) Let $M$ be a smooth manifold. If $V$ is a finite-
dimensional Lie subalgebra of $\mathfrak{X}(M)$, then the distribution
$\mathcal{D}^{V}$ is integrable.

One relevant problem concerns the determination of the form of the strata of a
stratification. If a stratification is regular on an open subset $U\subset M$,
then one can define locally a set of functionally independent functions whose
level sets are the strata of the stratification. Indeed, this amounts to the
integration of a standard distribution. If a stratification is singular, the
previous statement is not longer true and other methods are needed. The
following facts are useful to understand further parts of this work. First,
given a point $x\in M$ and a Vessiot–Guldberg Lie algebra $V$ on $M$, the
stratum $\mathcal{F}_{k}$ of the generalised distribution $\mathcal{D}^{V}$
passing through $x$ is given by the points of the form [48]

$\exp(X_{1})\circ\exp(X_{2})\circ\ldots\circ\exp(X_{s})x,$ (3.1)

where $s$ is any natural number, $X_{1},\ldots,X_{s}$ are any vector fields of
$V$, and each $\exp(X)$ stands for the local diffeomorphism on $M$ induced by
the vector field $X$ on $M$. Let $f$ be a function on $M$ such that if $X\in
V$, then there exists a function $f_{X}\in C^{\infty}(M)$ such that
$Xf=f_{X}f$. This can be considered as a not necessarily polynomial analogue
of a Darboux function for a vector field. Let us assume that $f=0$ at a point
of $\mathcal{F}_{k}$. It immediately follows from (3.1) that $f$ vanishes on
the whole $\mathcal{F}_{k}$. Hence, two points $x_{1},x_{2}\in M$ such that
$f(x_{1})\neq f(x_{2})$ cannot belong to the same $\mathcal{F}_{k}$. As shown
in following sections, $f$ does not need to be a constant of motion of the
vector fields in $V$. This is specially relevant when determining the strata
of an integrable singular generalised foliation, as their strata cannot always
be determined locally as the zeroes of a family of common first-integrals of
the vector fields of $V$. Meanwhile, this can always be achieved locally for
the leaves of an integrable distribution [1, 51].

##  4 Obtaining a matrix representation for a Lie algebra with nontrival
center

If a Lie algebra $\mathfrak{g}$ has a nontrivial center, then the image of the
map ${\rm ad}:\mathfrak{g}\ni v\mapsto[v,\cdot]\in\mathfrak{gl}(\mathfrak{g})$
is a matrix Lie algebra that is not isomorphic to $\mathfrak{g}$. Hence, the
image of ad does not give a matrix algebra representation of $\mathfrak{g}$,
which may give rise to a problem as matrix Lie algebra representation are
quite practical in computations [5]. The aim of this section is to provide a
method to obtain a matrix Lie algebra isomorphic to $\mathfrak{g}$ when its
center, $\mathcal{Z}(\mathfrak{g})$, is not trivial, namely
$\mathcal{Z}(\mathfrak{g})\neq 0$, and $\mathfrak{g}$ satisfy some additional
conditions. Our method is to be employed in the rest of our work during
calculations.

Let $\\{e_{1},\ldots,e_{s},e_{s+1},\ldots,e_{n}\\}$ be a basis of
$\mathfrak{g}$ such that $\\{e_{1},\ldots,e_{s}\\}$ form a basis for
$\mathcal{Z}(\mathfrak{g})$. Let $c_{ij}^{k}$, with $i,j,k=1,\ldots,n$, be the
structure constants of $\mathfrak{g}$ in the given basis, i.e.
$[e_{i},e_{j}]=\sum_{k=1}^{n}c_{ij}^{k}e_{k}$ for $i,j=1,\ldots,n$. Define
then $\tilde{\mathfrak{g}}:=\langle e_{1},\ldots,e_{n},e\rangle$. Let us
determine the conditions ensuring that $\tilde{\mathfrak{g}}$ is a Lie algebra
relative to a Lie bracket, $[\cdot,\cdot]_{\bar{\mathfrak{g}}}$, that is an
extension of the Lie bracket on $\mathfrak{g}$, i.e.
$[e^{\prime},e^{\prime\prime}]_{\bar{\mathfrak{g}}}=[e^{\prime},e^{\prime\prime}]_{\mathfrak{g}}$
for every $e^{\prime},e^{\prime\prime}\in\mathfrak{g}$, and
$[e,e_{i}]_{\bar{\mathfrak{g}}}=\alpha_{i}e_{i}$ for $i=1,\ldots,n$ so that

$\alpha_{1}\cdot\alpha_{2}\cdot\ldots\cdot\alpha_{s}\neq 0.$ (4.1)

The meaning of the last condition will be clear in a while. If
$[\cdot,\cdot]_{\bar{\mathfrak{g}}}$ is to be a Lie bracket, its Jacobi
identity leads to the following conditions

$\begin{gathered}{}_{\bar{\mathfrak{g}}}]_{\bar{\mathfrak{g}}}=[[e,e_{i}]_{\bar{\mathfrak{g}}},e_{j}]_{\bar{\mathfrak{g}}}+[e_{i},[e,e_{j}]_{\bar{\mathfrak{g}}}]_{\bar{\mathfrak{g}}}\iff\sum_{k=1}^{n}\alpha_{k}c_{ij}^{k}e_{k}=\sum_{k=1}^{n}(\alpha_{i}+\alpha_{j})c_{ij}^{k}e_{k},\qquad\forall
i,j=1,\ldots,n,\\\
[e,[e,e_{j}]_{\bar{\mathfrak{g}}}]_{\bar{\mathfrak{g}}}=[[e,e]_{\bar{\mathfrak{g}}},e_{j}]_{\bar{\mathfrak{g}}}+[e,[e,e_{j}]_{\bar{\mathfrak{g}}}]_{\bar{\mathfrak{g}}}=[e,[e,e_{j}]_{\bar{\mathfrak{g}}}]_{\bar{\mathfrak{g}}},\qquad\forall
j=1,\ldots,n.\end{gathered}$ (4.2)

Thus, previous conditions can be reduced to requiring that, for those indices
$i,j,k$ satisfying $c_{ij}^{k}\neq 0$, one gets

$\alpha_{i}+\alpha_{j}=\alpha_{k}.$

If the latter condition is satisfied, a new Lie algebra $\tilde{\mathfrak{g}}$
arises. Note that $\mathfrak{g}$ is an ideal of $\tilde{\mathfrak{g}}$. This
leads to a Lie algebra morphism
$\mathscr{R}:v\in{\mathfrak{g}}\mapsto\mathscr{R}_{v}:=[v,\cdot]_{\tilde{\mathfrak{g}}}\in\mathfrak{gl}(\tilde{\mathfrak{g}})$.
If $\mathscr{R}_{v}=0$, then $[v,e_{i}]_{\tilde{\mathfrak{g}}}=0$ for
$i=1,\ldots,n$, which means that $v\in\mathcal{Z}(\mathfrak{g})$. Thus,
$v=\sum_{i=1}^{s}\lambda_{i}e_{i}$ for certain constants
$\lambda_{1},\ldots,\lambda_{s}$ and
$0=[e,v]_{\tilde{\mathfrak{g}}}=\sum_{i=1}^{s}\lambda_{i}\alpha_{i}e_{i}$.
Then, condition (4.1) yields $\lambda_{1}=\ldots=\lambda_{s}=0$ and it turns
out that $v=0$. Hence, the elements $\mathscr{R}_{v}$, with
$v\in\mathfrak{g}$, span a matrix Lie algebra of endomorphisms on
$\tilde{\mathfrak{g}}$ isomorphic to $\mathfrak{g}$.

It is clear that conditions (4.1) and (4.2) do not need to be satisfied for a
general Lie algebra $\mathfrak{g}$ with a nontrivial center. Nevertheless, we
used in this work that this method works for all indecomposable real four-
dimensional Lie algebras with non-trivial center (see [50] for a complete list
of such Lie algebras). This will be enough for our purposes.

Let us illustrate our method. Consider the Lie algebra
$\mathfrak{s}_{1}=\langle e_{1},e_{2},e_{3},e_{4}\rangle$ with non-vanishing
commutation relations $[e_{4},e_{2}]=e_{1},[e_{4},e_{3}]=e_{3}$. Let us
construct a new Lie algebra $\tilde{\mathfrak{g}}=\langle
e_{1},e_{2},e_{3},e_{4},e\rangle$ following (4.1) and (4.2). This gives rise
to the following system of equations:

$\alpha_{4}+\alpha_{2}=\alpha_{1},\quad\alpha_{4}+\alpha_{3}=\alpha_{3},\quad\alpha_{1}\neq
0.$

We have that $\mathcal{Z}(\mathfrak{s}_{1})=\langle e_{1}\rangle$. The
previous system, under the corresponding restriction (4.1), has a solution
$\alpha_{4}=0,\alpha_{3}\in\mathbb{R},\alpha_{2}=\alpha_{1},\alpha_{1}\in\mathbb{R}\backslash\\{0\\}$.
In particular, set $\alpha_{2}=\alpha_{1}=1$ and $\alpha_{3}=0$. Thus, the
endomorphisms $\mathscr{R}_{e_{i}}$ on $\tilde{\mathfrak{g}}$ read

$\mathscr{R}_{e_{1}}=\left(\begin{array}[]{ccccc}0&0&0&0&-1\\\ 0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right),\mathscr{R}_{e_{2}}=\left(\begin{array}[]{ccccc}0&0&0&-1&0\\\
0&0&0&0&-1\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right),\mathscr{R}_{e_{3}}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&0&0&0&0\\\ 0&0&0&-1&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right),\mathscr{R}_{e_{4}}=\left(\begin{array}[]{ccccc}0&1&0&0&0\\\
0&0&0&0&0\\\ 0&0&1&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right).$

As desired, the above matrices have the same non-vanishing commutation
relations as $\mathfrak{s}_{1}$, i.e.

$[\mathscr{R}_{e_{4}},\mathscr{R}_{e_{2}}]=\mathscr{R}_{e_{1}},\qquad[\mathscr{R}_{e_{4}},\mathscr{R}_{e_{3}}]=\mathscr{R}_{e_{3}}.$

Let us prove that not every Lie algebra $\mathfrak{g}$ with non-trivial center
admits an extended Lie algebra $\tilde{\mathfrak{g}}$ of the form required by
our method. Consider the Lie algebra $\mathfrak{s}_{6,231}=\langle
e_{1},\ldots,e_{6}\rangle$ with nonzero commutation relations (see [50] for
details):

$[e_{2},e_{3}]=e_{1},\quad[e_{5},e_{1}]=e_{1},\quad[e_{5},e_{2}]=e_{2},\quad[e_{6},e_{1}]=e_{1},\quad[e_{6},e_{3}]=e_{3},\quad[e_{6},e_{5}]=e_{4}.$

Thus, $\mathcal{Z}(\mathfrak{s}_{4,231})=\langle e_{4}\rangle$. The
corresponding system of equations (4.2) reads

$\alpha_{2}+\alpha_{3}=\alpha_{1},\quad\alpha_{5}+\alpha_{1}=\alpha_{1},\quad\alpha_{5}+\alpha_{2}=\alpha_{2},\quad\alpha_{6}+\alpha_{1}=\alpha_{1},\quad\alpha_{6}+\alpha_{3}=\alpha_{3},\quad\alpha_{6}+\alpha_{5}=\alpha_{4}.$

Then, $\alpha_{5}=0,\alpha_{6}=0$ and $\alpha_{4}=\alpha_{5}=0$, which
contradicts the assumption of our method (4.2), namely $\alpha_{4}\neq 0$.

Consider the 7-dimensional Lie algebra $\mathfrak{g}:=\langle
e_{1},\ldots,e_{7}\rangle$ with $\mathcal{Z}(\mathfrak{g})=\langle
e_{7}\rangle$ and nonzero commutation relations (cf. [47, p. 492, case
$7_{I}$])

$\begin{gathered}=e_{3},\quad[e_{1},e_{3}]=e_{4},\quad[e_{1},e_{4}]=e_{5},\quad[e_{1},e_{6}]=e_{7},\quad[e_{2},e_{3}]=e_{6},\\\
[e_{2},e_{4}]=e_{7},\quad[e_{2},e_{5}]=e_{7},\quad[e_{2},e_{6}]=e_{7},\quad[e_{3},e_{4}]=-e_{7}.\end{gathered}$

The corresponding system (4.2) reads

$\begin{array}[]{rlrlrl}{\rm(i)}&\alpha_{1}+\alpha_{2}=\alpha_{3},&{\rm(ii)}&\alpha_{1}+\alpha_{3}=\alpha_{4},&{\rm(iii)}&\alpha_{1}+\alpha_{4}=\alpha_{5},\\\
{\rm(iv)}&\alpha_{1}+\alpha_{6}=\alpha_{7},&{\rm(v)}&\alpha_{2}+\alpha_{3}=\alpha_{6},&{\rm(vi)}&\alpha_{2}+\alpha_{4}=\alpha_{7},\\\
{\rm(vii)}&\alpha_{2}+\alpha_{5}=\alpha_{7},&{\rm(viii)}&\alpha_{2}+\alpha_{6}=\alpha_{7},&{\rm(ix)}&\alpha_{3}+\alpha_{4}=\alpha_{7}.\end{array}$

From (vii) and (viii), we get $\alpha_{5}=\alpha_{6}$; from (vi) and (ix), we
get $\alpha_{2}=\alpha_{3}$; and from (iv) and (viii), we get
$\alpha_{1}=\alpha_{2}$. Thus, $\alpha_{1}=\alpha_{2}=\alpha_{3}$. From (i),
it follows that $\alpha_{1}=0$. Thus, $\alpha_{2}=\alpha_{3}=0$. From (ii), we
get $\alpha_{4}=0$ and (iii) yields $\alpha_{5}=0$. From (iv), one gets
$\alpha_{6}=\alpha_{7}$. From (ix), we get $\alpha_{7}=0$, and thus
$\alpha_{6}=0$. Therefore, $\alpha_{1}=\ldots=\alpha_{7}=0$ is the only
possibility, but it does not satisfy (4.1).

##  5 Darboux families and the Lie algebra $\mathfrak{s}_{1}$

Let us classify solutions to the mCYBE for the Lie algebra $\mathfrak{s}_{1}$
up to its Lie algebra automorphisms. As a byproduct, we introduce a
generalisation of the concept of Darboux polynomial of a vector field.

The space of derivations on $\mathfrak{s}_{1}$ can straightforwardly be
obtained. It is indeed the solution of a linear algebra problem that can be
easily solved by hand calculation and/or via any mathematical computation
program. The same could be achieved for any four, five, or even some other
higher-dimensional Lie algebra. In particular, derivations on
$\mathfrak{s}_{1}$ take the form

$\mathfrak{der}(\mathfrak{s}_{1})=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&0&\mu_{14}\\\
0&\mu_{11}&0&\mu_{24}\\\ 0&0&\mu_{33}&\mu_{34}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{33},\mu_{14},\mu_{24},\mu_{34}\in\mathbb{R}\right\\}$
(5.1)

in the basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$ appearing in Table 1. By
Proposition 2.2 and, more specifically, Remark 2.1, the Lie algebra
$V_{\mathfrak{s}_{1}}$ of fundamental vector fields of the natural Lie group
action of ${\rm Aut}(\mathfrak{s}_{1})$ on $\Lambda^{2}\mathfrak{s}_{1}$ is
spanned by the basis (over the reals)

$\begin{gathered}X_{1}=2x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}}+x_{4}\partial_{x_{4}}+x_{5}\partial_{x_{5}},\qquad
X_{2}=x_{4}\partial_{x_{2}}+x_{5}\partial_{x_{3}},\qquad
X_{3}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\\\
X_{4}=x_{3}\partial_{x_{1}}-x_{6}\partial_{x_{4}},\qquad
X_{5}=x_{2}\partial_{x_{2}}+x_{4}\partial_{x_{4}}+x_{6}\partial_{x_{6}},\qquad
X_{6}=x_{3}\partial_{x_{2}}+x_{5}\partial_{x_{4}}.\end{gathered}$ (5.2)

As $X_{1},\ldots,X_{6}$ close on a finite-dimensional Lie algebra, Theorem 3.3
shows that they span an integrable generalised distribution
$\mathcal{D}^{V_{\mathfrak{s}_{1}}}$ on $\Lambda^{2}\mathfrak{s}_{1}$. In view
of (3.1) the strata of $\mathcal{D}^{V_{\mathfrak{s}_{1}}}$ are the orbits of
the action of the connected part of the identity of ${\rm Aut}(\mathfrak{g})$,
let us say ${\rm Aut}_{c}(\mathfrak{s}_{1})$, on
$\Lambda^{2}\mathfrak{s}_{1}$.

We define $M(p)$ to be a matrix whose entry $(i,j)$ is the $j$-coefficient of
$X_{i}$ at $p\in\Lambda^{2}\mathfrak{s}_{1}$ in the basis
$\partial_{x_{1}}|_{p},\ldots,\partial_{x_{6}}|_{p}$, namely

$M(p):=\left(\begin{array}[]{cccccc}2x_{1}&x_{2}&x_{3}&x_{4}&x_{5}&0\\\
0&x_{4}&x_{5}&0&0&0\\\ -x_{5}&-x_{6}&0&0&0&0\\\ x_{3}&0&0&-x_{6}&0&0\\\
0&x_{2}&0&x_{4}&0&x_{6}\\\ 0&x_{3}&0&x_{5}&0&0\end{array}\right),\qquad
p:=(x_{1},\ldots,x_{6})\in\Lambda^{2}\mathfrak{s}_{1}.$

The rank of $\mathcal{D}^{V_{\mathfrak{s}_{1}}}$ at
$p\in\Lambda^{2}\mathfrak{s}_{1}$ is equal to the rank of $M(p)$. It is simple
to calculate that the mCYBE for $\mathfrak{s}_{1}$ is given by the common
zeroes of the functions on $\Lambda^{2}\mathfrak{s}_{1}$ of the form

$f_{1}:=x_{3}x_{4},\quad f_{2}:=x_{3}x_{6},\quad f_{3}:=x_{5}^{2}.$ (5.3)

It is immediate that the space of solutions to the mCYBE, let us say
$\mathcal{Y}_{\mathfrak{s}_{1}}$, is not a submanifold in
$\Lambda^{2}\mathfrak{s}_{1}$. Nevertheless,

$X_{i}f_{j}=\sum_{k=1}^{3}\lambda_{ij}^{k}f_{k},\qquad i=1,\ldots,6,\quad
j=1,2,3,$ (5.4)

for certain constants $\lambda_{ij}^{k}$, with $i=1,\ldots,6$ and $j,k=1,2,3$.
Relations (5.4) show that the integral curves of any vector field in
$V_{\mathfrak{s}_{1}}$ passing through $w\in\mathcal{Y}_{\mathfrak{s}_{1}}$ is
contained in $\mathcal{Y}_{\mathfrak{s}_{1}}$. In fact, the derivative of the
functions $f_{1},f_{2},f_{3}$ along an integral curve, $\Psi(t)$, of a vector
field $X_{j}\in V_{\mathfrak{s}_{1}}$ such that
$\Psi(0)\in\mathcal{Y}_{\mathfrak{s}_{1}}$ reads

$\frac{df_{i}(\Psi(t))}{dt}=(X_{j}f_{i})(\Psi(t))=\sum_{k=1}^{3}\lambda_{ji}^{k}f_{k}(\Psi(t)),\qquad
i=1,2,3.$

Hence, the values of the $f_{i}(\Psi(t))$, with $i=1,2,3$, can be understood
as the solutions to a linear system of first-order ordinary differential
equations in normal form with constant coefficients and zero initial condition
since $f_{i}(\Psi(0))=0$ for $i=1,2,3$. Hence, $f_{1},f_{2},f_{3}$ vanish
along $\Psi(t)$ and, since (3.1) shows that the integral curves for all the
vector fields in $V$ connect all the points within the same strata of
$\mathcal{D}^{V_{\mathfrak{s}_{1}}}$, one obtains that the functions
$f_{1},f_{2},f_{3}$ are zero on any strata of
$\mathcal{D}^{V_{\mathfrak{s}_{1}}}$ containing a point where
$f_{1},f_{2},f_{3}$ vanish. It is worth noting that $f_{1},f_{2},f_{3}$ are
not constants of motion common to all the vector fields of
$V_{\mathfrak{s}_{1}}$. We hereafter call $\langle f_{1},f_{2},f_{3}\rangle$ a
Darboux family for the Lie algebra of vector fields $\langle
X_{1},\ldots,X_{6}\rangle$ on $\Lambda^{2}\mathfrak{s}_{1}$. More generally,
we propose the following definition and we extend previous results to a more
general realm.

######  Definition 5.1.

We call an $s$-dimensional linear space of functions $\mathcal{A}:=\langle
f_{1},\ldots,f_{s}\rangle$ on a manifold $M$ a Darboux family for a
Vessiot–Guldberg Lie algebra $V$ on $M$ if, for every $X\in V$ and $f_{j}$,
with $j=1,\ldots,s$, one can write

$Xf_{j}=\sum_{i=1}^{s}h_{jX}^{i}f_{i},$

for a certain family of smooth functions, $h_{jX}^{i}$, with $i=1,\ldots,s$,
on $M$, the so-called co-factors of $f_{j}$ relative to $X$ and the basis
$\\{f_{1},\ldots,f_{s}\\}$. The subset $\mathcal{S}_{\mathcal{A}}:=\\{p\in
M:f(p)=0,\forall f\in\mathcal{A}\\}$ is called the locus of the Darboux family
$\mathcal{A}$.

It is worth stressing that we require the functions $f_{1},\ldots,f_{s}$ and
$h^{i}_{jX}$, for $i,j=1,\ldots,s$ and $X\in V$, in the above definition to be
smooth. If all the functions $h_{jX}^{i}$ are equal to zero for every $X\in V$
and $i=1,\ldots,s$, then $f_{j}$ becomes a constant of motion for the vector
fields of $V$ on $M$. If the $h_{jX}^{i}$ are constants for every $X\in V$ and
$i,j=1,\ldots,s$, then we say that the Darboux family is linear. In this case,
$V$ gives rise to a Lie algebra representation $\rho:X\in V\mapsto
D_{X}\in{\rm End}(\mathcal{A})$, where $D_{X}$ stands for the action of the
vector field $X$ on the space of functions $\mathcal{A}$.

The vector fields of $V$ span a generalised distribution, which is integrable
and leads to a stratification of $M$ by some disjoint immersed submanifolds
which may have different dimensions. Darboux families are interesting to us
because the integral curves of vector fields in $V$ passing through the set
$\mathcal{S}_{\mathcal{A}}$ of common zeroes for the elements of $\mathcal{A}$
remain within it. Let us prove and analyse this fact, which represents a
rather simple generalisation of the argument given for $\mathfrak{s}_{1}$.
Consider a point in the locus $\mathcal{S}_{\mathcal{A}}$ and consider a basis
$Y_{1},\ldots,Y_{q}$ of the Lie algebra $V$ of vector fields. Consider an
integral curve $\Psi(t)$ of a vector field of $V$ passing through a point
$\Psi(0)\in\mathcal{S}_{\mathcal{A}}$. Then, the time derivative of the
functions $f_{1}(\Psi(t)),\ldots,f_{s}(\Psi(t))$ is

$\frac{df_{i}(\Psi(t))}{dt}=(Xf_{i})\Psi(t)=\sum_{i=1}^{s}h_{iX}^{j}(\Psi(t))f_{j}(\Psi(t)),\qquad
i=1,\ldots,s.$

Hence, the values of the $f_{i}(\Psi(t))$ can be understood as the solutions
to the linear system of first-order ordinary differential equations in normal
form with $t$-dependent coefficients

$\frac{du_{i}}{dt}=\sum_{j=1}^{s}h^{j}_{iX}(\Psi(t))u_{j},\qquad
i=1,\ldots,s,$

with initial condition $u_{1}=\ldots=u_{s}=0$. By the existence and uniqueness
theorem, the solution to the previous Cauchy problem is
$u_{1}(t)=\ldots=u_{s}(t)=0$. Therefore, the functions $f_{1},\ldots,f_{s}$
vanish on the integral curves of $X$. From this and the decomposition (3.1),
we can infer that the functions $f_{1},\ldots,f_{s}$ vanish on the strata of
the distribution $\mathcal{D}^{V}$ containing a point within
$\mathcal{S}_{\mathcal{A}}$. In other words, the strata of $\mathcal{D}^{V}$
containing some point of $\mathcal{S}_{A}$ are fully contained in
$\mathcal{S}_{A}$.

In view of the above, Darboux families can be used to reduce the determination
of the strata in $M$ of the generalised distribution $\mathcal{D}^{V}$ to
determining the strata within ${\mathcal{S}_{\mathcal{A}}}$ and out of
${\mathcal{S}_{\mathcal{A}}}$. This will be specially interesting to obtain
the strata of $\mathcal{D}^{V}$ at points where the generalised distribution
is not regular and it may not exist a constant of motion common to all the
vector fields in $V$ that can be used to obtain the strata of
$\mathcal{D}^{V}$. Note also that if $\langle f_{1},\ldots,f_{s}\rangle$ form
a Darboux family for $V$, then the set of common zeroes of $\langle
f_{1},\ldots,f_{s}\rangle$ must be the sum (as subsets of $M$) of a collection
of strata of the generalised distribution $\mathcal{D}^{V}$.

Let us study how to use Darboux families to study solutions to mCYBEs and
CYBEs. Let us start by a general result.

######  Theorem 5.2.

Let $\mathcal{A}^{(3)}_{\mathfrak{g}}$ be a Darboux family for
$V_{\mathfrak{g}}^{(3)}$ on $\Lambda^{3}\mathfrak{g}$. Then, the space of
functions

$\mathcal{A}:=\left\\{g\in
C^{\infty}(\Lambda^{2}\mathfrak{g}):\,g=f\circ[\cdot,\cdot],\,f\in\mathcal{A}_{\mathfrak{g}}^{(3)}\right\\}$
(5.5)

is a Darboux family for $V_{\mathfrak{g}}$ on $\Lambda^{2}\mathfrak{g}$. If
$\mathcal{A}_{\mathfrak{g}}^{(3)}$ is a linear Darboux family, then
$\mathcal{A}$ is also a linear Darboux family.

###### Proof.

Since each $X^{(2)}\in V_{\mathfrak{g}}$ is a fundamental field of the action
of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$, its flow is given
by the one-parametric group of diffeomorphisms $\Lambda^{2}T_{t}$ induced by a
certain one-parametric group of Lie algebra automorphisms $T_{t}\in{\rm
Aut}(\mathfrak{g})$, $t\in\mathbb{R}$. Every $g\in\mathcal{A}$ is of the form
$g(r)=f([r,r])$ for some $f\in\mathcal{A}_{\mathfrak{g}}^{(3)}$ and every
$r\in\Lambda^{2}\mathfrak{g}$. Then,

$(X^{(2)}g)(r)=\frac{d}{dt}\bigg{|}_{t=0}g(\Lambda^{2}T_{t}(r))=\frac{d}{dt}\bigg{|}_{t=0}f([\Lambda^{2}T_{t}(r),\Lambda^{2}T_{t}(r)])\\\
=\frac{d}{dt}\bigg{|}_{t=0}f(\Lambda^{3}T_{t}[r,r])=(X^{(3)}f)([r,r])=\sum_{i=1}^{r}h^{i}([r,r])f_{i}([r,r]),$

where $f_{1},\ldots,f_{s}$ form a basis of the Darboux family
$\mathcal{A}_{\mathfrak{g}}^{(3)}$, the functions $h^{1},\ldots,h^{s}$ are the
cofactors of $f$ relative to $X^{(3)}$ and the basis
$\\{f_{1},\ldots,f_{s}\\}$, and $X^{(3)}$ is the fundamental vector field of
the action of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{3}\mathfrak{g}$ induced
by the one-parameter group $\\{T_{t}\\}_{t\in\mathbb{R}}$ of Lie algebra
automorphisms of $\mathfrak{g}$. Since $f_{1}([r,r]),\ldots,f_{s}([r,r])$
belong to $\mathcal{A}$ and $h^{1}([r,r]),\ldots,h^{s}([r,r])$ are functions
on $\Lambda^{2}\mathfrak{g}$, the $\mathcal{A}$ forms a Darboux family for
$V_{\mathfrak{g}}$ on $\Lambda^{2}\mathfrak{g}$.

If $\mathcal{A}^{(3)}_{\mathfrak{g}}$ is a linear Darboux family, then
$h^{1},\ldots,h^{s}$ are constants. Therefore, the cofactors
$h^{1}([r,r]),\ldots,h^{s}([r,r])$ are also constants, which makes
$\mathcal{A}$ into a linear Darboux family for $V_{\mathfrak{g}}$ on
$\Lambda^{2}\mathfrak{g}$. ∎

In short, the following proposition shows that the solutions to the mCYBE on
$\mathfrak{g}$ can be considered as the locus of a Darboux family relative to
$V_{\mathfrak{g}}$. A similar result can be applied to the CYBE on any Lie
algebra $\mathfrak{g}$.

######  Proposition 5.1.

Let $((\Lambda^{3}\mathfrak{g})^{\mathfrak{g}})^{\circ}$ be the annihilator of
$(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$, i.e. the subspace of elements of
$(\Lambda^{3}\mathfrak{g})^{*}$ vanishing on
$(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$. The functions

$f_{\upsilon}:r\in\Lambda^{2}\mathfrak{g}\mapsto\upsilon([r,r])\in\mathbb{R},\qquad\upsilon\in((\Lambda^{3}\mathfrak{g})^{\mathfrak{g}})^{\circ},$
(5.6)

span a linear Darboux family, $\mathcal{A}$, for the Lie algebra
$V_{\mathfrak{g}}$ of fundamental vector fields of the action of ${\rm
Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$. Its locus is the set of
solutions to the mCYBE for $\mathfrak{g}$. Moreover, the components of the
CYBE in a coordinate system given by a basis of $\Lambda^{2}\mathfrak{g}^{*}$
span also a linear Darboux family for $V_{\mathfrak{g}}$ on
$\Lambda^{2}\mathfrak{g}$.

###### Proof.

Let us prove that the annihilator of
$(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$ is a Darboux family for
$V_{\mathfrak{g}}^{(3)}$ on $\Lambda^{3}\mathfrak{g}$. In fact, if $X^{(3)}\in
V_{\mathfrak{g}}^{(3)}$, then its flow is given by a one-parameter group of
diffeomorphisms of the form $\Lambda^{3}T_{t}$ for certain Lie algebra
automorphisms $T_{t}\in{\rm Aut}(\mathfrak{g})$ with $t\in\mathbb{R}$.
Consider a function $f\in\Lambda^{3}\mathfrak{g}^{*}$ such that $f(w)=0$ for
every $w\in(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$, i.e.
$f\in((\Lambda^{3}\mathfrak{g})^{\mathfrak{g}})^{\circ}$. Then,

$(X^{(3)}f)(w)=\frac{d}{dt}\bigg{|}_{t=0}f((\Lambda^{3}T_{t})(w))=\frac{d}{dt}\bigg{|}_{t=0}f\left(w_{t}\right)=0,$

where we have used that $(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$ is closed
under the extension to $\Lambda^{3}\mathfrak{g}$ of Lie algebra automorphisms
of $\mathfrak{g}$ and thus
$(\Lambda^{3}T_{t})(w)=:w_{t}\in(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$.
Hence, $X^{(3)}f$ vanishes on $(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$.
Since $X^{(3)}$ is a linear vector field on $\Lambda^{3}\mathfrak{g}$ and $f$
is a linear function on $\Lambda^{3}\mathfrak{g}$, one obtains that $X^{(3)}f$
is a linear function on $\Lambda^{3}\mathfrak{g}$ vanishing on
$(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$. Hence, $X^{(3)}f$ is a linear
combination of elements of a basis of
$((\Lambda^{3}\mathfrak{g})^{\mathfrak{g}})^{\circ}$ and
$((\Lambda^{3}\mathfrak{g})^{\mathfrak{g}})^{\circ}$ becomes a linear Darboux
family of $V_{\mathfrak{g}}^{(3)}$ on $\Lambda^{3}\mathfrak{g}$. By Theorem
5.2, the space of functions (5.6) becomes a linear Darboux family for
$V_{\mathfrak{g}}$ on $\Lambda^{2}\mathfrak{g}$. The locus of (5.6) is the
space of solutions of the mCYBE for $\mathfrak{g}$.

Since every $X^{(3)}\in V^{(3)}_{\mathfrak{g}}$ is linear vector field on a
linear coordinate system on $\Lambda^{3}\mathfrak{g}$, the space
$\Lambda^{3}\mathfrak{g}^{*}$ is also a Darboux family for
$V_{\mathfrak{g}}^{(3)}$ on $\Lambda^{3}\mathfrak{g}$. By Theorem 5.2, one has
that the induced by (5.5) family of functions on $\Lambda^{2}\mathfrak{g}$ is
a linear Darboux family relative to $V_{\mathfrak{g}}$ on
$\Lambda^{2}\mathfrak{g}$. The Darboux family is indeed spanned by the
components of the CYBE of $\mathfrak{g}$, i.e. $[r,r]=0$, and its locus is its
set of solutions. ∎

A natural question is how to determine Darboux families. Here, we provide
several results about them that will be useful to as so as to derive new
Darboux families from known ones.

######  Proposition 5.2.

If $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are Darboux families for the same
Vessiot–Guldberg Lie algebra $V$ on a manifold $M$, then the sum
$\mathcal{A}_{1}+\mathcal{A}_{2}$ is a Darboux family for $V$ on $M$ as well.

The proof of the previous result is immediate. Obviously, the locus of
$\mathcal{A}_{1}+\mathcal{A}_{2}$ is contained in the locus of
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$. This will be interesting in next
sections to distinguish between different strata of the generalised
distribution spanned by the vector fields of $V_{\mathfrak{g}}$ on
$\mathcal{Y}_{\mathfrak{g}}$.

In next sections, it will be relevant to study the linear Darboux families for
a certain $V_{\mathfrak{g}}$ given by a one-dimensional
$\mathcal{A}\subset(\Lambda^{2}\mathfrak{g})^{*}$. This is due to the fact
that, as seen in Section 7, the sum of such Darboux families as vector spaces
will generate new useful Darboux families. We will call such a linear one-
dimensional Darboux family a brick. Bricks are one-dimensional representations
of $V_{\mathfrak{g}}$ on $\Lambda^{2}\mathfrak{g}^{*}$.

Note that, for every linear Darboux family for $V_{\mathfrak{g}}$, bricks are
easy to obtain since, in view of (2.3), they are given by the intersection of
an eigenvector space for each endomorphism of the form
$\Lambda^{2}d\in\mathfrak{gl}(\Lambda^{2}\mathfrak{g})$ with
$d\in\mathfrak{der}(\mathfrak{g})$.

Let us give another interesting proposition, which is an immediate consequence
of the fact that the vector fields of $V_{\mathfrak{g}}$ are linear relative
to a linear coordinate set on $\Lambda^{2}\mathfrak{g}$.

######  Proposition 5.3.

The space $\Lambda^{2}\mathfrak{g}^{*}$ is a linear Darboux family of
functions on $\Lambda^{2}\mathfrak{g}$ relative to the Lie algebra
$V_{\mathfrak{g}}$.

Since $\Lambda^{2}\mathfrak{g}^{*}$ is a linear Darboux family of
$V_{\mathfrak{g}}$, there exists a linear representation of
$\mathfrak{aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}^{*}$. Its
irreducible representations also give rise to Darboux families, which can be
summed (as linear spaces) to the Darboux family, or potentially elements
thereof, associated with the mCYBE to determine new Darboux families. The
locus of such sums will be interested to as when they contain elements of
$\mathcal{Y}_{\mathfrak{g}}$. This will be employed to obtain the strata of
$V_{\mathfrak{g}}$ within $\mathcal{Y}_{\mathfrak{g}}$ and, therefore,
inequivalent $r$-matrices relative to the action of ${\rm
Aut}_{c}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$. This in turn will be
used to obtain families of inequivalent $r$-matrices and coboundary
cocommutators for real four-dimensional indecomposable Lie algebras [50] in
Section 7.

##  6 Geometric structure of solutions to mCYBEs and Darboux families

Let us study several details on the geometry of the space of solutions to
modified and non-modified classical Yang-Baxter equations, the problem of
classification up to Lie algebra automorphisms of coboundary Lie bialgebras,
and Darboux families. As previously, we hereafter write $V_{\mathfrak{g}}$ for
the Lie algebra of fundamental vector fields of the action of ${\rm
Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$, whilst
$\mathscr{E}_{\mathfrak{g}}$ stands for the distribution on
$\Lambda^{2}\mathfrak{g}$ spanned by the vector fields of $V_{\mathfrak{g}}$.
We recall that we have defined
$\Lambda^{2}_{R}\mathfrak{g}:=\Lambda^{2}\mathfrak{g}/(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}$
and
$\pi_{\mathfrak{g}}:w\in\Lambda^{2}\mathfrak{g}\mapsto[w]\in\Lambda^{2}_{R}\mathfrak{g}$
stands for the canonical projection onto the quotient space
$\Lambda_{R}^{2}\mathfrak{g}$. As previously, we hereafter write
$\mathcal{Y}_{\mathfrak{g}}$ for the space of $r$-matrices of $\mathfrak{g}$.

The action of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$ induces
an action of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}_{R}\mathfrak{g}$ of the
form (see [35, Lemma 8.3] for details)

$\Psi:T\in{\rm
Aut}(\mathfrak{g})\mapsto[\Lambda^{2}T]\in{GL}(\Lambda^{2}_{R}\mathfrak{g}),\qquad[\Lambda^{2}T]([w]):=[\Lambda^{2}T(w)],\qquad\forall
w\in\Lambda^{2}\mathfrak{g}.$

The above result leads to the following consequence

$\pi_{\mathfrak{g}}((\Lambda^{2}T)(w))=[\Psi(T)](\pi_{\mathfrak{g}}(w)),\qquad\forall
w\in\Lambda^{2}\mathfrak{g},\qquad\forall T\in{\rm Aut}(\mathfrak{g}).$ (6.1)

Recall that two actions of a Lie group $G$, let us say $\Phi_{i}:G\times
N_{i}\rightarrow N_{i}$ with $i=1,2$, are equivariant relative to
$\phi:N_{1}\rightarrow N_{2}$ if $\phi\circ\Phi_{1}(g,x)=\Phi_{2}(g,\phi(x))$
for every $g\in G$ and $x\in N_{1}$. It can be proved that if $\Phi_{1}$ and
$\Phi_{2}$ are equivariant, the Lie algebra of fundamental vector fields of
$\Phi_{1}$ projects, via $\phi_{*}$, onto the Lie algebra of fundamental
vector fields of $\Phi_{2}$. Moreover, the orbits of $\Phi_{1}$ project, via
$\phi$, onto the orbits of $\Phi_{2}$ (see [1] for details).

Expression (6.1) shows that the actions of ${\rm Aut}(\mathfrak{g})$ on
$\Lambda^{2}\mathfrak{g}$ and $\Lambda^{2}_{R}\mathfrak{g}$ are equivariant
relative to
$\pi_{\mathfrak{g}}:\Lambda^{2}\mathfrak{g}\rightarrow\Lambda^{2}_{R}\mathfrak{g}$.
Then, the vector fields of $V_{\mathfrak{g}}$ project via
$\pi_{\mathfrak{g}*}$ onto the Lie algebra of fundamental vector fields of the
action of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}_{R}\mathfrak{g}$. We write
$\mathscr{E}^{R}_{\mathfrak{g}}$ for the generalised distribution spanned by
the fundamental vector fields of the action of ${\rm Aut}(\mathfrak{g})$ on
$\Lambda^{2}_{R}\mathfrak{g}$. This means that the strata of the generalised
distribution $\mathscr{E}_{\mathfrak{g}}$ project onto the strata of the
generalised distribution $\mathscr{E}^{R}_{\mathfrak{g}}$.

As the elements of ${\rm Aut}(\mathfrak{g})$ map solutions to the mCYBE of
$\mathfrak{g}$ onto solutions of the same equation, the orbits of ${\rm
Aut}(\mathfrak{g})$ containing a solution to the mCYBE consist of solutions to
the mCYBE. Similarly, the orbits of the action of ${\rm Aut}(\mathfrak{g})$ on
$\Lambda^{2}\mathfrak{g}$ containing a solution to the CYBE consist of
solutions to the CYBE. Two $r$-matrices are equivalent up to Lie algebra
automorphisms of $\mathfrak{g}$ if and only if they belong to the same orbit
of ${\rm Aut}(\mathfrak{g})$ within $\mathcal{Y}_{\mathfrak{g}}$.

Recall that ${\rm Aut}(\mathfrak{g})/{\rm Aut}_{c}(\mathfrak{g})$, where ${\rm
Aut}_{c}(\mathfrak{g})$ is the connected component of the identity of the Lie
group ${\rm Aut}(\mathfrak{g})$, is discrete and therefore countable [13]. In
other words, the Lie group ${\rm Aut}(\mathfrak{g})$ is a numerable sum (as
subsets) of disjoint and connected subsets of ${\rm Aut}(\mathfrak{g})$
diffeomorphic to ${\rm Aut}_{c}(\mathfrak{g})$. The strata of
$\mathscr{E}_{\mathfrak{g}}$ coincide with the orbits of ${\rm
Aut}_{c}(\mathfrak{g})$. Therefore, the orbits of ${\rm Aut}(\mathfrak{g})$
are given by a numerable sum of strata of $\mathscr{E}_{\mathfrak{g}}$.
Moreover, each particular orbit of ${\rm Aut}(\mathfrak{g})$ is an immersed
submanifold in $\Lambda^{2}\mathfrak{g}$ of a fixed dimension. Hence, each one
of the orbits of ${\rm Aut}_{c}(\mathfrak{g})$, whose sum gives rise to an
orbit of ${\rm Aut}(\mathfrak{g})$, must have the same dimension. Similarly,
the orbits of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}_{R}\mathfrak{g}$ are
given by a numerable sum (as subsets) of strata of
$\mathscr{E}^{R}_{\mathfrak{g}}$ of the same dimension, which are orbits of
the action of ${\rm Aut}_{c}(\mathfrak{g})$ on $\Lambda^{2}_{R}\mathfrak{g}$.

It can be proved that two coboundary cocommutators can be equivalent up to Lie
algebra automorphisms even when their associated $r$-matrices are not. For
instance, the zero cocommutator
$\delta_{0}:\mathfrak{g}\rightarrow\mathfrak{g}\wedge\mathfrak{g}$ can take
the form $0=\delta_{0}=[r,\cdot]$ for $r=0$ or for any other
$r_{1}\in(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}\backslash\\{0\\}$.
Nevertheless, the $0$ and $r_{1}$ cannot be connected by the action of an
element of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$, since the
action of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$ is linear and
$0\in\Lambda^{2}\mathfrak{g}$ is an orbit. In spite of that, the cocommutators
generated by $0$ and $r_{1}$ are the same. Let us give the conditions ensuring
that two coboundary cocommutators on $\mathfrak{g}$ are equivalent up to a Lie
algebra automorphism of $\mathfrak{g}$.

######  Proposition 6.1.

Two coboundary cocommutators
$\delta_{i}:v\in\mathfrak{g}\mapsto[r_{i},v]\in\mathfrak{g}\wedge\mathfrak{g}$,
with $i=1,2$ and $r_{i}\in\mathcal{Y}_{\mathfrak{g}}$, are equivalent under a
Lie algebra automorphism of $\mathfrak{g}$ if and only if there exists
$T\in{\rm Aut}(\mathfrak{g})$ such that $(\Lambda^{2}T)(r_{1})$ and $r_{2}$
belong to the same equivalence class in $\Lambda^{2}_{R}\mathfrak{g}$.

###### Proof.

It is clear that the transformation of $\delta_{1}$ by $T$ reads, by the
properties of the Schouten bracket, as follows

$\Lambda^{2}T\circ\delta_{1}\circ
T^{-1}=\Lambda^{2}T\circ[r_{1},T^{-1}(\cdot)]=[\Lambda^{2}Tr_{1},\cdot].$

Hence,

$\delta_{2}=\Lambda^{2}T\circ\delta_{1}\circ
T^{-1}\Leftrightarrow[\Lambda^{2}Tr_{1},\cdot]=[r_{2},\cdot]\Leftrightarrow[\Lambda^{2}Tr_{1}-r_{2},\cdot]=0,$

and the last condition amounts to the fact that
$\Lambda^{2}Tr_{1}-r_{2}\in(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}$, which
implies that $\Lambda^{2}Tr_{1},r_{2}$ belong to the same equivalence class in
$\Lambda_{R}^{2}\mathfrak{g}$. ∎

Note that two $r$-matrices are equivalent (relative to the action of an
element of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$) if and only
if they belong to the same family of strata of the distribution
$\mathscr{E}_{\mathfrak{g}}$ giving rise to an orbit of ${\rm
Aut}(\mathfrak{g})$ in $\mathcal{Y}_{\mathfrak{g}}$. Nevertheless, two
$r$-matrices give rise to two equivalent coboundary cocommutators if and only
if they belong to the same orbit of ${\rm Aut}(\mathfrak{g})$ on
$\Lambda^{2}_{R}\mathfrak{g}$. In other words, we have proven the following
proposition.

######  Proposition 6.2.

There exists a one-to one correspondence between the $r$-matrices for
$\mathfrak{g}$ that are equivalent up to an element of ${\rm
Aut}_{c}(\mathfrak{g})$ and the strata of the generalised distribution
$\mathscr{E}_{\mathfrak{g}}$ within $\mathcal{Y}_{\mathfrak{g}}$. Moreover,
there exists a one-to-one correspondence between the families of equivalent
(up to Lie algebra automorphisms of $\mathfrak{g}$) coboundary cocommutators
of a Lie algebra $\mathfrak{g}$ and the orbits of the action of ${\rm
Aut}(\mathfrak{g})$ on
$\pi_{\mathfrak{g}}(\mathcal{Y}_{\mathfrak{g}})\subset\Lambda^{2}_{R}\mathfrak{g}$.
Every such an orbit in $\pi_{\mathfrak{g}}(\mathcal{Y}_{\mathfrak{g}})$ is the
sum (as subsets) of a numerable collection of strata of the same dimension of
the generalised distribution $\mathscr{E}_{\mathfrak{g}}^{R}$.

In practice, we shall obtain each orbit of ${\rm Aut}(\mathfrak{g})$ on
$\mathcal{Y}_{\mathfrak{g}}$ as a numerable family of strata of
$\mathscr{E}_{\mathfrak{g}}$ in $\mathcal{Y}_{\mathfrak{g}}$. Such orbits
represent the families of $r$-matrices that are equivalent up to an element of
${\rm Aut}(\mathfrak{g})$. Then, we will derive all strata in
$\mathcal{Y}_{\mathfrak{g}}$ that map onto the same space in
$\Lambda_{R}^{2}\mathfrak{g}$. This last task will give us, along with the
orbits of ${\rm Aut}(\mathfrak{g})$ on $\mathcal{Y}_{\mathfrak{g}}$, the
equivalent classes of coboundary cocommutators on $\mathfrak{g}$ up to Lie
algebra automorphisms of $\mathfrak{g}$.

Let us study how Darboux families can be employed to obtain the orbits of
${\rm Aut}(\mathfrak{g})$ on $\Lambda^{2}\mathfrak{g}$ and
$\Lambda_{R}^{2}\mathfrak{g}$. In particular, Darboux families are employed
here to identify the strata of $\mathscr{E}_{\mathfrak{g}}$.

######  Proposition 6.3.

The locus, $\ell_{\mathfrak{g}}$, of a Darboux family
$\mathcal{A}^{E}_{\mathfrak{g}}$ relative to the Lie algebra
$V_{\mathfrak{g}}$ on $\Lambda^{2}\mathfrak{g}$ is a sum (as subsets) of the
orbits of the action of ${\rm Aut}_{c}(\mathfrak{g})$ on
$\Lambda^{2}\mathfrak{g}$ containing a point in $\ell_{\mathfrak{g}}$. If
$\ell_{\mathfrak{g}}$ is a connected submanifold in $\Lambda^{2}\mathfrak{g}$
of dimension given by the rank of the generalised distribution
$\mathscr{E}_{\mathfrak{g}}$, then $\ell_{\mathfrak{g}}$ is a strata of the
generalised distribution $\mathscr{E}_{\mathfrak{g}}$.

###### Proof.

For each point $p$ in $\ell_{\mathfrak{g}}$ the orbit $\mathcal{O}_{p}$ of
${\rm Aut}_{c}(\mathfrak{g})$ passing through $p$ has a tangent space spanned
by the vector fields of $V_{\mathfrak{g}}$. By the definition of a Darboux
family for $V_{\mathfrak{g}}$ and the fact that the functions of
$\mathcal{A}^{E}_{\mathfrak{g}}$ vanish at $p$, one has that all functions of
$\mathcal{A}^{E}_{\mathfrak{g}}$ are equal to zero on $\mathcal{O}_{p}$ (see
Section 5). Hence, $\mathcal{O}_{p}$ is contained in $\ell_{\mathfrak{g}}$.
Then, $\ell_{\mathfrak{g}}$ is the sum (as subsets) of orbits of ${\rm
Aut}_{c}(\mathfrak{g})$ passing through the points of $\ell_{\mathfrak{g}}$.

Recall that the strata of $\mathscr{E}_{\mathfrak{g}}$ are the orbits of ${\rm
Aut}_{c}(\mathfrak{g})$ acting on $\Lambda^{2}\mathfrak{g}$. Hence,
$\ell_{\mathfrak{g}}$ is a sum of strata of $\mathscr{E}_{\mathfrak{g}}$. If
the rank of $\mathscr{E}_{\mathfrak{g}}$ on $\ell_{\mathfrak{g}}$ is equal to
$\dim\ell_{\mathfrak{g}}$, where $\ell_{\mathfrak{g}}$ is assumed to be a
connected submanifold, then $\ell_{\mathfrak{g}}$ is locally generated around
any point $p\in\ell_{\mathfrak{g}}$ by the action of ${\rm
Aut}_{c}(\mathfrak{g})$ on that point. Since $\ell_{1}$ is connected, it is
wholly generated by the action of ${\rm Aut}_{c}(\mathfrak{g})$ and it becomes
a strata of $\mathscr{E}_{\mathfrak{g}}$. ∎

Note that we are interested only in those loci of Darboux families of
$V_{\mathfrak{g}}$ contained in the space $\mathcal{Y}_{\mathfrak{g}}$ of
solutions to the mCYBE of $\mathfrak{g}$.

It is interesting that necessary conditions for the existence of a Lie algebra
automorphism connecting two $r$-matrices can be given using
$\Lambda^{3}\mathfrak{g}$. Consider, for instance, the following proposition,
whose proof is straightforward.

######  Proposition 6.4.

If $r_{1}$ and $r_{2}$ are two equivalent $r$-matrices for a Lie algebra
$\mathfrak{g}$, then, $[r_{1},r_{1}]$ and $[r_{2},r_{2}]$ belong to the same
orbit of ${\rm Aut}(\mathfrak{g})$ acting on
$(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$.

The necessary condition in Proposition 6.4 is not sufficient. For instance, it
may happen that two $r$-matrices are solutions to the classical Yang-Baxter
equation without being equivalent (as seen in Table 2). From a practical point
of view, Proposition 6.4 shows that the determination of the orbits of the
action of ${\rm Aut}(\mathfrak{g})$ on $\Lambda^{3}\mathfrak{g}$ may be of
interest. Following the reasoning for studying the equivalence of $r$-matrices
in $\Lambda^{2}\mathfrak{g}$, the orbits of the action of ${\rm
Aut}(\mathfrak{g})$ on $\Lambda^{3}\mathfrak{g}$ can be obtained by obtaining
the strata of the distribution, $\mathscr{E}^{(3)}_{\mathfrak{g}}$, spanned by
the fundamental vector fields $V^{(3)}_{\mathfrak{g}}$ of the action of ${\rm
Aut}_{c}(\mathfrak{g})$ on $\Lambda^{3}\mathfrak{g}$. Such strata can be
obtained through Darboux families for $V^{(3)}_{\mathfrak{g}}$ in an analogous
way to the method employed to study $\mathscr{E}_{\mathfrak{g}}$. Then, the
action of one element of each connected component of ${\rm Aut}(\mathfrak{g})$
on the strata of $V^{(3)}_{\mathfrak{g}}$ in
$(\Lambda^{2}\mathfrak{g})^{\mathfrak{g}}$ gives the orbits of ${\rm
Aut}(\mathfrak{g})$ within $(\Lambda^{3}\mathfrak{g})^{\mathfrak{g}}$.

######  Proposition 6.5.

If two $r$-matrices $r_{1},r_{2}$ are equivalent, then the rank of
$r_{1},r_{2}$, as bilinear mappings on $\mathfrak{g}^{*}$, are the same.

###### Proof.

Recall that $r$-matrices can be considered as the antisymmetric bilinear
mappings on $\mathfrak{g}^{*}$ associated with them. The action of ${\rm
Aut}(\mathfrak{g})$ on an $r$-matrix does not change its rank as a bilinear
antisymmetric mapping. Hence, equivalent $r$-matrices must have the same rank.
∎

Although $r$-matrices whose bilinear antisymmetric mappings on
$\mathfrak{g}^{*}$ have different rank are not equivalent up to Lie algebra
automorphisms of $\mathfrak{g}$, it may happen that $r$-matrices related to
the same rank are not equivalent. In fact, although there always exists in
this latter case an $A\in GL(\mathfrak{g})$ acting on
$\Lambda^{2}\mathfrak{g}$ mapping one element of this space into another of
the same order ($A$ can be obtained by mapping both bivectors into a canonical
form), $A$ does not necessarily belong to ${\rm Aut}(\mathfrak{g})$. Hence,
both bivectors need not be equivalent up to Lie algebra automorphisms of
$\mathfrak{g}$. Many examples of this will appear in Table 2.

##  7 Classification of coboundary Lie bialgebras on four-dimensional
indecomposable Lie algebras

A Lie algebra $\mathfrak{g}$ is decomposable when it can be written as the
direct sum of two of its proper ideals. Otherwise, we say that $\mathfrak{g}$
is indecomposable. Winternitz and Šnobl classified in [50, Part 4] all
indecomposable Lie algebras up to dimension six (see [50, p. 217] for comments
on previous classifications) . The aim of this section is to present the
classification up to Lie algebra automorphisms of coboundary Lie bialgebras on
real four-dimensional indecomposable Lie algebras via Darboux families and the
Winternitz–Šnobl (WŠ) classification. Moreover, the equivalence of
$r$-matrices and solutions to CYBEs for four-dimensional real indecomposable
Lie algebras are also analysed.

We hereafter assume $\mathfrak{g}$ to be a real four-dimensional and
indecomposable Lie algebra. For the sake of completeness, the structure
constants for every $\mathfrak{g}$ in a basis $\\{e_{1},e_{2},e_{3},e_{4}\\}$
are given in Table 1. We hereafter endow $\Lambda^{2}\mathfrak{g}$ with a
basis $\\{e_{12}:=e_{1}\wedge e_{2},e_{13}:=e_{1}\wedge
e_{3},e_{14}:=e_{1}\wedge e_{4},e_{23}:=e_{2}\wedge e_{3},e_{24}:=e_{2}\wedge
e_{4},e_{34}:=e_{3}\wedge e_{4}\\}$, while its dual basis is denoted by
$\\{x_{1},x_{2},x_{3},x_{4},x_{5},x_{6}\\}$. Additionally,
$\Lambda^{3}\mathfrak{g}$ is endowed with a basis
$\\{e_{123},e_{124},e_{134},e_{234}\\}$. Moreover, due to the bilinearity and
symmetry of the Schouten bracket on $\Lambda^{2}\mathfrak{g}$, it is enough to
know $[e_{ij},e_{kl}]$, with $1\leq i<k\leq 4$ or
$i=k\in\\{1,\ldots,4\\},1\leq j\leq l\leq 4$, to determine the Schouten
bracket on the whole $\Lambda^{2}\mathfrak{g}$. In Tables 3–5, we summarise
several relevant Schouten brackets between the basis elements of
$\mathfrak{g}$, $\Lambda^{2}\mathfrak{g}$, and $\Lambda^{3}\mathfrak{g}$ that
can easily be obtained via Table 1 and, eventually, by means of a symbolic
computation program. Anyway, we add such calculations to easily follow the
proofs of our following results. The classes of Lie algebras in Table 1 may
contain several non-isomorphic Lie algebras depending on several parameters,
e.g. $\alpha,\beta$. To simplify the notation, we will not detail specific
values of the parameters when we refer to properties of a whole class of Lie
algebras and/or it is clear the particular case we are discussing from the
context.

Lie algebra | $[e_{1},e_{2}]$ | $[e_{1},e_{3}]$ | $[e_{1},e_{4}]$ | $[e_{2},e_{3}]$ | $[e_{2},e_{4}]$ | $[e_{3},e_{4}]$ | Parameters
---|---|---|---|---|---|---|---
$\mathfrak{s}_{1}$ | 0 | 0 | 0 | 0 | $-e_{1}$ | $-e_{3}$ |
$\mathfrak{s}_{2}$ | 0 | 0 | $-e_{1}$ | 0 | $-e_{1}-e_{2}$ | $-e_{2}-e_{3}$ |
$\mathfrak{s}_{3}$ | 0 | 0 | $-e_{1}$ | 0 | $-\alpha e_{2}$ | $-\beta e_{3}$ |  $0<|\beta|\leq|\alpha|\leq 1$,
$(\alpha,\beta)\neq(-1,-1)$
$\mathfrak{s}_{4}$ | 0 | 0 | $-e_{1}$ | 0 | $-e_{1}-e_{2}$ | $-\alpha e_{3}$ | $\alpha\in\mathbb{R}\backslash\\{0\\}$
$\mathfrak{s}_{5}$ | 0 | 0 | $-\alpha e_{1}$ | 0 | $-\beta e_{2}+e_{3}$ | $-e_{2}-\beta e_{3}$ | $\alpha>0,\beta\in\mathbb{R}$
$\mathfrak{s}_{6}$ | 0 | 0 | 0 | $e_{1}$ | $-e_{2}$ | $e_{3}$ |
$\mathfrak{s}_{7}$ | 0 | 0 | 0 | $e_{1}$ | $e_{3}$ | $-e_{2}$ |
$\mathfrak{s}_{8}$ | 0 | 0 | $-(1+\alpha)e_{1}$ | $e_{1}$ | $-e_{2}$ | $-\alpha e_{3}$ | $\alpha\in]-1,1]\backslash\\{0\\}$
$\mathfrak{s}_{9}$ | 0 | 0 | $-2\alpha e_{1}$ | $e_{1}$ | $-\alpha e_{2}+e_{3}$ | $-e_{2}-\alpha e_{3}$ | $\alpha>0$
$\mathfrak{s}_{10}$ | 0 | 0 | $-2e_{1}$ | $e_{1}$ | $-e_{2}$ | $-e_{2}-e_{3}$ |
$\mathfrak{s}_{11}$ | 0 | 0 | $-e_{1}$ | $e_{1}$ | $-e_{2}$ | 0 |
$\mathfrak{s}_{12}$ | 0 | $-e_{1}$ | $e_{2}$ | $-e_{2}$ | $-e_{1}$ | 0 |
$\mathfrak{n}_{1}$ | 0 | 0 | 0 | 0 | $e_{1}$ | $e_{2}$ |
Table 1: Structure constants for real four-dimensional indecomposable Lie
algebras according to the WŠ classification. Such Lie algebras are denoted by
$\mathfrak{s}_{4,k},\mathfrak{n}_{4,1}$ for $1\leq k\leq 12$ in the WŠ
classification. Nevertheless, the subindex concerning the dimension of the Lie
algebra will be removed in our work to simplify the notation. The letters
$\mathfrak{n}$ and $\mathfrak{s}$ are used to denote nilpotent and solvable
(but not nilpotent) Lie algebras, respectively. To avoid duplicities of Lie
algebras, the parameters $\alpha,\beta$ of $\mathfrak{s}_{3}$ must satisfy
additional restrictions that were not detailed explicitly in [50] (see [50, p.
228]). Such restrictions will be determined in Subsection 7.3.

It is interesting to discuss some physical and mathematical applications of
the Lie bialgebras given in our list and their relations to some previous
works. For instance, the nilpotent Lie algebra $\mathfrak{n}_{1}$ corresponds
to the so-called Galilean Lie algebra, which is deformed in the study of XYZ
models with the quantum Galilei group appearing in [12]. Lie bialgebras
defined on Lie algebras of the type $\mathfrak{n}_{1}$ also occur in [42, 43].
The coboundary Lie bialgebras on the Lie algebra $\mathfrak{s}_{6}$, which
corresponds to the so-called harmonic oscillator algebra, are obtained in [6]
without studying the equivalence up to Lie algebra automorphisms. In that
work, the quantisations and $R$-matrices of such a sort of Lie bialgebras are
also studied. In particular, equation (3.7) in [6] matches the mCYBE obtained
in Section 7.6. The work [7] proves that all Lie bialgebras on the harmonic
oscillator algebra are coboundary ones. This shows that our classification of
Lie bialgebras on $\mathfrak{s}_{6}$ finishes the classification of all Lie
bialgebras on the harmonic oscillator algebra. On the other hand, the Lie
algebra $\mathfrak{s}_{8}$ with $\alpha=-1/2$ is isomorphic to the Lie algebra
$A\bar{G}_{2}(1)$ in [42]. The Lie algebra $\mathfrak{s}_{7}$ provides a
central extension of Caley-Klein Lie algebras analysed in [10, eq. (3.13)].
Other cases of real four-dimensional indecomposable Lie bialgebras can be
found in [4, 44], where, for instance, two-dimensional central extended
Galilei algebras are studied.

###  7.1 Lie algebra $\mathfrak{s}_{1}$

The structure constants for Lie algebra $\mathfrak{s}_{1}$ are given in Table
1, while relevant Schouten brackets between the elements of the bases of
$\mathfrak{s}_{1}$, $\Lambda^{2}\mathfrak{s}_{1}$,
$\Lambda^{3}\mathfrak{s}_{1}$ to be used hereafter are displayed in Tables
3–5. In particular, one sees from such tables that
$(\Lambda^{2}\mathfrak{s}_{1})^{\mathfrak{s}_{1}}=\langle e_{12}\rangle$ and
$(\Lambda^{3}\mathfrak{s}_{1})^{\mathfrak{s}_{1}}=0$.

To obtain the classification of $r$-matrices for $\mathfrak{s}_{1}$ up to Lie
algebra automorphisms thereof, we will rely on the determination of the strata
of the distribution $\mathscr{E}_{\mathfrak{s}_{1}}$ (see Proposition 6.2) via
Darboux families for $V_{\mathfrak{s}_{1}}$ and a few elements of ${\rm
Aut}(\mathfrak{s}_{1})$.

By Proposition 6.3, the locus of a Darboux family for a Lie algebra
$V_{\mathfrak{s}_{1}}$ is the sum (as subsets) of strata of the generalised
distribution $\mathscr{E}_{\mathfrak{s}_{1}}$ spanned by the vector fields of
$V_{\mathfrak{s}_{1}}$. In particular, we are interested in those loci being a
submanifold within $\mathcal{Y}_{\mathfrak{s}_{1}}$ of dimension matching the
rank of $\mathscr{E}_{\mathfrak{\mathfrak{s}_{1}}}$ on them. This is due to
the fact that, in view of Proposition 6.3, their connected components are the
orbits of ${\rm Aut}_{c}(\mathfrak{s}_{1})$ on
$\mathcal{Y}_{\mathfrak{s}_{1}}$. With this aim, we derive the fundamental
vector fields of the action of ${\rm Aut}(\mathfrak{s}_{1})$ on
$\Lambda^{2}\mathfrak{s}_{1}$. By Remark 2.1, the derivations of
$\mathfrak{s}_{1}$ take the form (5.1), and a basis of $V_{\mathfrak{s}_{1}}$
is given by (5.2).

For an element $r\in\Lambda^{2}\mathfrak{s}_{1}$, we get

$[r,r]=2(-x_{2}x_{5}+x_{3}x_{4}-x_{4}x_{5})e_{123}-2x_{5}^{2}e_{124}+2(x_{3}-x_{5})x_{6}e_{134}+2x_{5}x_{6}e_{234}.$

This expression can easily be derived by using the properties of the algebraic
Schouten bracket and the structure constants for $\mathfrak{s}_{1}$. A similar
computation would not be difficult even for many higher-dimensional Lie
algebras.

Since $(\Lambda^{3}\mathfrak{s}_{1})^{\mathfrak{s}_{1}}=0$, the mCYBE and the
CYBE for $\mathfrak{s}_{1}$ are equal and read

$x_{3}x_{4}=0,\quad x_{3}x_{6}=0,\quad x_{5}=0.$

It is interesting that, since
$(\Lambda^{3}\mathfrak{s}_{2})^{\mathfrak{s}_{1}}=0$, every $r$-matrix for
$\mathfrak{s}_{1}$ amounts to a left-invariant Poisson bivector on
$\mathfrak{s}_{1}^{*}$ (cf. [15]).

Let us apply to $\mathfrak{s}_{1}$ the following procedure to obtain the
orbits of ${\rm Aut}_{c}(\mathfrak{g})$ on $\mathcal{Y}_{\mathfrak{g}}$ for a
Lie algebra $\mathfrak{g}$ of arbitrary dimension, i.e. the strata of
$\mathscr{E}_{\mathfrak{g}}$ within $\mathcal{Y}_{\mathfrak{g}}$. We start by
considering a locus $\ell_{1}$ of a Darboux family of $V_{\mathfrak{g}}$ on
$\Lambda^{2}\mathfrak{g}$. As we are interested in determining the strata of
$\mathscr{E}_{\mathfrak{g}}$ within $\mathcal{Y}_{\mathfrak{g}}$, we look for
an $\ell_{1}$ containing some solution to the mCYBE, namely
$\ell_{1}\cap\mathcal{Y}_{\mathfrak{g}}\neq\emptyset$. Recall that the strata
of $\mathscr{E}_{\mathfrak{g}}$ are completely contained within the locus or
completely contained off the locus of every Darboux family for
$V_{\mathfrak{g}}$. This divides the strata of $\mathscr{E}_{\mathfrak{g}}$ on
$\mathcal{Y}_{\mathfrak{g}}$ into those within or outside $\ell_{1}$. If
$\ell_{1}\cap\mathcal{Y}_{\mathfrak{g}}$ is a submanifold of dimension given
by the rank of $\mathscr{E}_{\mathfrak{g}}$ on it, its connected parts are the
strata of $\mathscr{E}_{\mathfrak{g}}$ in
$\ell_{1}\cap\mathcal{Y}_{\mathfrak{g}}$. Otherwise, to obtain the strata
within $\ell_{1}\cap\mathcal{Y}_{\mathfrak{g}}$, we look for a new Darboux
family for $V_{\mathfrak{g}}$ containing the previous Darboux family so that
its new locus, let us say $\ell_{2}$, will satisfy
$\ell_{2}\cap\mathcal{Y}_{\mathfrak{g}}\neq\emptyset$. This process is
repeated until we obtain a locus, $\ell_{k}$, that is a submanifold of
dimension equal to the rank of $\mathscr{E}_{\mathfrak{g}}$ on it. In other
words, the connected components of the locus $\ell_{k}$ are the orbits of
${\rm Aut}_{c}(\mathfrak{g})$ in $\ell_{k}$.

To determine all the strata of $\mathscr{E}_{\mathfrak{g}}$ on
$\mathcal{Y}_{\mathfrak{g}}\cap(\ell_{k})^{c}$, where $(\ell_{k})^{c}$ stands
for the complementary subset of $\ell_{k}$ in $\Lambda^{2}\mathfrak{g}$, which
is always open, one considers a subspace of the final Darboux family and the
previous procedure is applied again iteratively to the obtained Darboux family
on $\Lambda^{2}\mathfrak{g}\backslash\ell_{k}$. At the end, we obtain another
family of strata of $\mathscr{E}_{\mathfrak{g}}$ and the procedure can be
repeated to search for remaining strata of $\mathscr{E}_{\mathfrak{g}}$ in
$\mathcal{Y}_{\mathfrak{g}}$. The above-described process, for all four-
dimensional indecomposable Lie algebras, allows us to obtain the orbits of
${\rm Aut}_{c}(\mathfrak{g})$ on $\mathcal{Y}_{\mathfrak{g}}$.

To represent schematically our procedure, we use a diagram, which is hereafter
called a Darboux tree. Every Darboux family of the above-mentioned method is
described by a set of boxes going from an edge of the diagram on the left to
one of the edges on the right. The squared boxes of the type $f=0$, for a
certain function $f$, give the generating functions of the Darboux family
while the squared boxes of the type $f\neq 0$, give the conditions restricting
the manifold in $\Lambda^{2}\mathfrak{g}$ where the Darboux family and
$V_{\mathfrak{g}}$ are restricted to. The connected parts of the loci of all
the Darboux families represented in a Darboux tree give rise to a
decomposition of $\mathcal{Y}_{\mathfrak{g}}$ into orbits of ${\rm
Aut}_{c}(\mathfrak{g})$. To help understanding the calculations, some oval
boxes with additional information, e.g. explaining why other possibilities are
not considered, are given. It is worth noting that bricks where employed to
generate Darboux families. In particular, the Darboux tree of
$\mathfrak{s}_{1}$ is given below. In this case, it is immediate that $x_{5}$
and $x_{6}$ are bricks for $V_{\mathfrak{s}_{1}}$. We use this fact to
generate Darboux families of the following Darboux tree.

$x_{5}=0$$x_{6}=0$$x_{6}\neq 0$$x_{3}=0$$x_{3}\neq
0$$x_{3}=0$$\stackrel{{\scriptstyle\tiny{\rm No\,solutions}}}{{x_{3}\neq
0}}$$x_{4}=0$$x_{4}\neq 0$$\stackrel{{\scriptstyle\tiny{\rm
No\,solutions}}}{{x_{4}\neq 0}}$$x_{4}=0$$x_{1}=0$$x_{1}\neq
0$$x_{2}=0$$x_{2}\neq 0$$x_{1}=0$$x_{1}\neq 0$$x_{1}=0$$x_{1}\neq
0$$x_{1}=0$$x_{1}\neq 0$0IIIIIIIVVVIVIIVIII

The connected components of the loci of the Darboux families of the previous
Darboux tree give the orbits of ${\rm Aut}_{c}(\mathfrak{s}_{1})$ on
$\mathcal{Y}_{\mathfrak{s}_{1}}$. Results are given in Table 2. Let us study
the equivalence of $r$-matrices up to action of the whole ${\rm
Aut}(\mathfrak{s}_{1})$. To do so, we use that the group of automorphisms of
$\mathfrak{s}_{1}$ reads

${\rm
Aut}(\mathfrak{s}_{1})=\left\\{\left(\begin{array}[]{cccc}T_{2}^{2}&T^{2}_{1}&0&T^{4}_{1}\\\
0&T^{2}_{2}&0&T^{4}_{2}\\\ 0&0&T_{3}^{3}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T_{2}^{2},T_{3}^{3}\in\mathbb{R}\backslash\\{0\\},\,T^{2}_{1},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\right\\}.$

This Lie group is easy to be derived from the structure constants of
$\mathfrak{s}_{1}$. In reality, it is enough for our purposes to consider an
element of each connected component of ${\rm Aut}(\mathfrak{s}_{1})$. It is
remarkable that Lie algebra automorphism groups can rather easily be obtained
for Lie algebras of relatively high dimension and/or satisfying special
properties, e.g. for semisimple Lie algebras [40]. In our case, it is
immediate that ${\rm Aut}(\mathfrak{s}_{1})$ has four such components. One
element of each connected component of ${\rm Aut}(\mathfrak{s}_{1})$ and its
extension to $\Lambda^{2}\mathfrak{s}_{1}$ are given by

${\small
T_{\lambda_{1},\lambda_{2}}:=\left(\begin{array}[]{cccc}\lambda_{1}&0&0&0\\\
0&\lambda_{1}&0&0\\\ 0&0&\lambda_{2}&0\\\
0&0&0&1\end{array}\right)\Longrightarrow\Lambda^{2}T_{\lambda_{1},\lambda_{2}}:=\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\
0&\lambda_{1}\lambda_{2}&0&0&0&0\\\ 0&0&\lambda_{1}&0&0&0\\\
0&0&0&\lambda_{1}\lambda_{2}&0&0\\\ 0&0&0&0&\lambda_{1}&0\\\
0&0&0&0&0&\lambda_{2}\\\
\end{array}\right),\qquad\lambda_{1},\lambda_{2}\in\\{\pm 1\\}}.$ (7.1)

The orbits of ${\rm Aut}(\mathfrak{s}_{1})$ in
$\mathcal{Y}_{\mathfrak{s}_{1}}$ are given by the action of
$\Lambda^{2}T_{\lambda_{1},\lambda_{2}}$ on the orbits of ${\rm
Aut}_{c}(\mathfrak{s}_{1})$ on $\mathcal{Y}_{\mathfrak{s}_{1}}$. By using all
mappings $\Lambda^{2}T_{\lambda_{1},\lambda_{2}}$, we can verify whether some
of the orbits of ${\rm Aut}_{c}(\mathfrak{s}_{1})$ on
$\mathcal{Y}_{\mathfrak{s}_{1}}$ can be connected among themselves by a Lie
algebra automorphism of $\mathfrak{s}_{1}$. Our results are summarised in
Table 2.

Recall that $(\Lambda^{2}\mathfrak{s}_{1})^{\mathfrak{s}_{1}}=\langle
e_{12}\rangle$. Consequently, all orbits of ${\rm Aut}(\mathfrak{s}_{1})$ on
$\mathcal{Y}_{\mathfrak{s}_{1}}$ mapping onto the same space in
$\Lambda^{2}_{R}\mathfrak{s}_{1}$ via $\pi_{\mathfrak{s}_{1}}$ give equivalent
coboundary coproducts up to the action of elements of ${\rm
Aut}(\mathfrak{s}_{1})$. In particular, we get five classes of coboundary
coproducts induced by the following classes of $r$-matrices: a) ${\rm
I}_{\pm}$, b) II, III±, c) IV, V±, d) VI, and e) VII, VIII±. We recall that
all given $r$-matrices are solutions to the CYBE.

###  7.2 Lie algebra $\mathfrak{s}_{2}$

Let us apply the formalism given in the previous section to Lie algebra
$\mathfrak{s}_{2}$. Lie algebra $\mathfrak{s}_{2}$ has structure constants
given in Table 1. Relevant Schouten brackets between the bases of elements of
$\mathfrak{s}_{2}$, $\Lambda^{2}\mathfrak{s}_{2}$, and
$\Lambda^{3}\mathfrak{s}_{2}$ are given in Tables 3–5. Remarkably,
$(\Lambda^{2}\mathfrak{s}_{2})^{\mathfrak{s}_{2}}=0$ and
$(\Lambda^{3}\mathfrak{s}_{2})^{\mathfrak{s}_{2}}=0$.

By Remark 2.1, it follows that

$\mathfrak{der}(\mathfrak{s}_{2})=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&\mu_{13}&\mu_{14}\\\
0&\mu_{11}&\mu_{12}&\mu_{24}\\\ 0&0&\mu_{11}&\mu_{34}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{13},\mu_{14},\mu_{24},\mu_{34}\in\mathbb{R}\right\\},$

which gives rise, by lifting these derivations to
$\Lambda^{2}\mathfrak{s}_{2}$, to the basis of $V_{\mathfrak{s}_{2}}$ of the
form

$\begin{gathered}X_{1}=2x_{1}\partial_{x_{1}}+2x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}}+2x_{4}\partial_{x_{4}}+x_{5}\partial_{x_{5}}+x_{6}\partial_{x_{6}},\quad
X_{2}=x_{2}\partial_{x_{1}}+x_{4}\partial_{x_{2}}+x_{5}\partial_{x_{3}}+x_{6}\partial_{x_{5}},\\\
X_{3}=-x_{4}\partial_{x_{1}}+x_{6}\partial_{x_{3}},\quad
X_{4}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\quad
X_{5}=x_{3}\partial_{x_{1}}-x_{6}\partial_{x_{4}},\quad
X_{6}=x_{3}\partial_{x_{2}}+x_{5}\partial_{x_{4}}.\end{gathered}$

Meanwhile, for $r\in\Lambda^{2}\mathfrak{s}_{2}$, one has

$[r,r]=2(2x_{1}x_{6}-2x_{2}x_{5}+x_{2}x_{6}+2x_{3}x_{4}-x_{4}x_{5})e_{123}+2(x_{3}x_{6}-x_{5}^{2})e_{124}-2x_{5}x_{6}e_{134}-2x_{6}^{2}e_{234}.$

Since $(\Lambda^{3}\mathfrak{s}_{2})^{\mathfrak{s}_{2}}=0$, the mCYBE and the
CYBE are the same and read

$x_{3}x_{4}=0,\quad x_{5}=0,\quad x_{6}=0.$

With the previous information, we are ready to obtain the classification of
orbits of the action of Aut${}_{c}(\mathfrak{s}_{2})$ on the subset of
$r$-matrices
$\mathcal{Y}_{\mathfrak{s}_{2}}\subset\Lambda^{2}\mathfrak{s}_{2}$ by using
Darboux families. We shall start by using bricks. The only brick of
$\mathfrak{s}_{2}$ is $x_{6}$. The procedure is accomplished as in the
previous section and it is summarised in the following Darboux tree.

$x_{6}=0$$x_{5}=0$$\stackrel{{\scriptstyle\tiny{\rm
No\,\,solutions}}}{{x_{5}\neq 0}}$$x_{4}=0$$x_{4}\neq 0$$x_{3}=0$$x_{3}\neq
0$$x_{3}=0$$\stackrel{{\scriptstyle\tiny{\rm No\,\,solutions}}}{{x_{3}\neq
0}}$$x_{2}=0$$x_{2}\neq 0$$x_{1}=0$$x_{1}\neq 0$0IIIIIIIV

As commented in Subsection 7.1, to obtain the orbits of ${\rm
Aut}(\mathfrak{s}_{2})$ on $\mathcal{Y}_{\mathfrak{s}_{2}}$, it is enough to
derive the extension to $\Lambda^{2}\mathfrak{s}_{2}$ of a single element of
each connected component of Aut$(\mathfrak{s}_{2})$. The automorphisms of the
Lie algebra $\mathfrak{s}_{2}$ can easily be obtained. They read

${\rm
Aut}(\mathfrak{s}_{2})=\left\\{\left(\begin{array}[]{cccc}T_{2}^{2}&T^{3}_{2}&T^{3}_{1}&T^{4}_{1}\\\
0&T^{2}_{2}&T^{3}_{2}&T^{4}_{2}\\\ 0&0&T^{2}_{2}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T^{2}_{2}\in\mathbb{R}\backslash\\{0\\},\,T_{1}^{3},T_{2}^{3},T_{1}^{4},T_{2}^{4},T_{3}^{4}\in\mathbb{R}\right\\}.$

Then, one element of ${\rm Aut}(\mathfrak{s}_{2})$ for each of its connected
components and their extensions to $\Lambda^{2}\mathfrak{s}_{2}$ read

$T_{\lambda}:=\left(\begin{array}[]{cccc}\lambda&0&0&0\\\ 0&\lambda&0&0\\\
0&0&\lambda&0\\\
0&0&0&1\end{array}\right)\Longrightarrow\Lambda^{2}T_{\lambda}={\small\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\
0&1&0&0&0&0\\\ 0&0&\lambda&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&\lambda&0\\\
0&0&0&0&0&\lambda\\\ \end{array}\right)},\qquad\lambda\in\\{\pm 1\\}.$ (7.2)

The nine connected parts of the submanifolds $0$, I±, II±, III, IV± are the
orbits of Aut${}_{c}(\mathfrak{s}_{2})$ on $\mathcal{Y}_{\mathfrak{s}_{2}}$.
It is simple to see which of them are further connected through an element of
Aut$(\mathfrak{s}_{2})$. In result, the orbits of Aut$(\mathfrak{s}_{2})$ on
$\mathcal{Y}_{\mathfrak{s}_{2}}$ are given by the eight submanifolds $0,{\rm
I}_{-},{\rm I}_{+},{\rm II}_{-},{\rm II}_{+},{\rm III},{\rm IV}_{-},{\rm
IV}_{+}$ given in Table 2. Since
$(\Lambda^{2}\mathfrak{s}_{2})^{\mathfrak{s}_{2}}=0$, each such a submanifold
gives a family of equivalent coboundary coproducts that is non-equivalent to
$r$-matrices within remaining submanifolds. This gives all possible classes of
non-equivalent coboundary coproducts. As in the previous subsection, all
$r$-matrices are solutions to the CYBE in $\mathfrak{s}_{2}$.

###  7.3 Lie algebra $\mathfrak{s}_{3}$

The structure constants of the Lie algebras of type $\mathfrak{s}_{3}$ are
given in Table 1. It was noted in [50, p. 228] that additional restrictions
must be imposed on the parameters $\alpha,\beta$ given in Table 1 to avoid the
repetition of isomorphic Lie algebras within the Lie algebra class
$\mathfrak{s}_{3}$. Such restrictions were not explicitly detailed in [50],
but it is immediate that isomorphic cases within Lie algebras of the class
$\mathfrak{s}_{3}$ can be classified by the adjoint action of $e_{4}$ on
$\langle e_{1},e_{2},e_{3}\rangle$. In particular, if $|\alpha|=|\beta|$, then
the Lie algebras with parameters $(\alpha,\beta)$ and $(\beta,\alpha)$ are
isomorphic relative to the Lie algebra isomorphism that interchanges $e_{2}$
with $e_{3}$ and leaves $e_{1}$ invariant. It will be relevant in what follows
that the lift of this Lie algebra automorphism to
$\Lambda^{2}\mathfrak{s}_{3}$ interchanges $x_{1}$ with $x_{2}$, $x_{5}$ with
$x_{6}$, it maps $x_{4}$ to $-x_{4}$, and it leaves $x_{3}$ invariant. Due to
the previous isomorphism, we restrict ourselves to the case $\alpha\geq\beta$
when $|\alpha|=|\beta|$.

Using the structure constants given in Table 1 and the induced Schouten
brackets between the relevant to our purposes basis elements of
$\mathfrak{s}_{3}$, $\Lambda^{2}\mathfrak{s}_{3}$, and
$\Lambda^{3}\mathfrak{s}_{3}$ in Tables 3–5, one gets that
$(\Lambda^{2}\mathfrak{s}_{3})^{\mathfrak{s}_{3}}\subset\\{e_{12},e_{13},e_{23}\\}$.
More specifically, $e_{12}\in(\Lambda^{2}\mathfrak{s}_{3})^{\mathfrak{s}_{3}}$
if and only if $\alpha=-1$; while
$e_{13}\in(\Lambda^{2}\mathfrak{s}_{3})^{\mathfrak{s}_{3}}$ if and only if
$\beta=-1$. Finally,
$e_{23}\in(\Lambda^{2}\mathfrak{s}_{3})^{\mathfrak{s}_{3}}$ just when
$\alpha+\beta=0$. Moreover,
$(\Lambda^{3}\mathfrak{s}_{3})^{\mathfrak{s}_{3}}=\langle e_{123}\rangle$ if
$\alpha+\beta=-1$ and $(\Lambda^{3}\mathfrak{s}_{3})^{\mathfrak{s}_{3}}=0$,
otherwise.

Since the Lie algebra class $\mathfrak{s}_{3}$ contains so many subcases that
relevant properties may change from one to another subcase with given
parameters $(\alpha,\beta)$, e.g. the Lie algebra automorphism group, we will
develop here a modification of our method consisting of applying the Darboux
family method to the fundamental vector fields of the action of the Lie group
${\rm Aut}_{\rm all}(\mathfrak{s}_{3})$ of all common Lie algebra
automorphisms for all parameters $\alpha,\beta$ and analysing its relation to
the Lie algebra automorphisms for each case,
$\mathfrak{s}_{3}^{\alpha,\beta}$, to obtain our final classification. To
simplify the notation, we skip the parameters $\alpha,\beta$ in
$\mathfrak{s}_{3}^{\alpha,\beta}$ when they are not needed to understand what
we are talking out, e.g. if we jest care about the dimension of the linear
space $\Lambda^{2}\mathfrak{s}_{3}$.

By Remark 2.1, one obtains the following common Lie algebra of derivations of
$\mathfrak{s}_{3}$ for all the values of $\alpha$ and $\beta$:

$\mathfrak{der}_{\rm
all}(\mathfrak{s}_{3})=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&0&0&\mu_{14}\\\
0&\mu_{22}&0&\mu_{24}\\\ 0&0&\mu_{33}&\mu_{34}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{22},\mu_{33},\mu_{14},\mu_{24},\mu_{34}\in\mathbb{R}\right\\},$

and, after lifting the elements of a basis of $\mathfrak{der}_{\rm
all}(\mathfrak{s}_{3})$ to $\Lambda^{2}\mathfrak{s}_{3}$, we obtain a basis of
$V^{\rm all}_{\mathfrak{s}_{3}}$, namely the Lie algebra of fundamental vector
fields of the action of the Lie algebra automorphism group ${\rm Aut}_{\rm
all}(\mathfrak{s}_{3})$ common to all $\alpha,\beta$ acting on
$\Lambda^{2}\mathfrak{s}_{3}$, of the form

$\begin{gathered}X_{1}=x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}},\quad
X_{2}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\quad
X_{3}=x_{1}\partial_{x_{1}}+x_{4}\partial_{x_{4}}+x_{5}\partial_{x_{5}},\\\
X_{4}=x_{3}\partial_{x_{1}}-x_{6}\partial_{x_{4}},\quad
X_{5}=x_{2}\partial_{x_{2}}+x_{4}\partial_{x_{4}}+x_{6}\partial_{x_{6}},\quad
X_{6}=x_{3}\partial_{x_{2}}+x_{5}\partial_{x_{4}}.\end{gathered}$

It is worth stressing that if $\alpha=\beta$ and/or one on the coefficients
$\alpha,\beta$ are equal to one, then the Lie algebra of derivations of the
particular $\mathfrak{s}^{\alpha,\beta}_{3}$ is larger due to the existence of
Lie algebra automorphisms of $\mathfrak{s}^{\alpha,\beta}_{3}$ leaving
invariant the eigenspaces of ${\rm ad}_{e_{4}}$ acting on $\langle
e_{1},e_{2},e_{3}\rangle$ (more detailed calculations can be seen at the end
of this subsection). By dealing with $\mathfrak{der}_{\rm
all}(\mathfrak{s}_{3})$, we shall derive the orbits of the connected part of
the identity of the group of common Lie algebra automorphisms for all
$\alpha,\beta$, i.e. ${\rm Aut}_{\rm all}(\mathfrak{s}_{3})$, on each
$\mathcal{Y}_{\mathfrak{s}^{\alpha,\beta}_{3}}$ via Darboux families. To
obtain the equivalence of $r$-matrices up to the action of ${\rm
Aut}(\mathfrak{s}^{\alpha,\beta}_{3})$ for each pair $(\alpha,\beta)$, we will
use the action of elements of ${\rm Aut}(\mathfrak{s}^{\alpha,\beta}_{3})$ not
contained in ${\rm Aut}_{\rm all,c}(\mathfrak{s}_{3})$ for each particular
pair of parameters $(\alpha,\beta)$.

For an element $r\in\Lambda^{2}\mathfrak{s}_{3}$, we get

$[r,r]=2[(1+\alpha)x_{1}x_{6}-(1+\beta)x_{2}x_{5}+(\alpha+\beta)x_{3}x_{4}]e_{123}+2(\alpha-1)x_{3}x_{5}e_{124}+2(\beta-1)x_{3}x_{6}e_{134}+2(\beta-\alpha)x_{5}x_{6}e_{234}.$

For $1+\alpha+\beta\neq 0$, we have
$(\Lambda^{3}\mathfrak{s}^{\alpha,\beta}_{3})^{\mathfrak{s}^{\alpha,\beta}_{3}}=0$.
Then, the mCYBE and the CYBE are the same in this case and they read

$(1+\alpha)x_{1}x_{6}-(1+\beta)x_{2}x_{5}+(\alpha+\beta)x_{3}x_{4}=0,\quad(\alpha-1)x_{3}x_{5}=0,\quad(\beta-1)x_{3}x_{6}=0,\quad(\beta-\alpha)x_{5}x_{6}=0.$

If $1+\alpha+\beta=0$, then
$(\Lambda^{3}\mathfrak{s}^{\alpha,-1-\alpha}_{3})^{\mathfrak{s}^{\alpha,-1-\alpha}_{3}}=\langle
e_{123}\rangle$ and the mCYBE reads

$(\alpha-1)x_{3}x_{5}=0,\quad(\beta-1)x_{3}x_{6}=0,\quad(\beta-\alpha)x_{5}x_{6}=0.$

Since $x_{3},x_{5},x_{6}$ are bricks of $\mathfrak{s}^{\alpha,\beta}_{3}$ for
every pair $(\alpha,\beta)$, our Darboux tree for the Darboux families of
$V_{\mathfrak{s}_{3}}^{\rm all}$ starts with cases $x_{i}=0$ and $x_{i}\neq
0$, $i\in\\{3,5,6\\}$. The full Darboux tree is presented below.

$x_{6}=0$$x_{5}=0$$x_{5}\neq 0$$x_{3}=0$$x_{3}\neq
0$$x_{3}=0$$\stackrel{{\scriptstyle\alpha=1}}{{x_{3}\neq
0}}$$x_{4}=0$$x_{4}\neq
0$$x_{4}=0$$\stackrel{{\scriptstyle\alpha+\beta\in\\{0,-1\\}}}{{x_{4}\neq
0}}$$x_{2}=0$$\stackrel{{\scriptstyle\beta=-1\,{\rm
or}\,\alpha+\beta=-1}}{{x_{2}\neq
0}}$$x_{3}x_{4}-x_{2}x_{5}=0$$\stackrel{{\scriptstyle\beta=-1}}{{x_{3}x_{4}-x_{2}x_{5}\neq
0}}$$x_{2}=0$$x_{2}\neq 0$$x_{2}=0$$x_{2}\neq 0$$x_{1}=0$$x_{1}\neq
0$$x_{1}=0$$x_{1}\neq 0$$x_{1}=0$$x_{1}\neq 0$$x_{1}=0$$x_{1}\neq
0$0IIIIIIIVVVIVIIVIII${\rm IX}_{\alpha+\beta\in\\{0,-1\\}}$X${\rm
XI}^{\beta=-1}_{\alpha+\beta=-1}$${\rm XII}_{\alpha=1}$${\rm
XIII}_{\alpha=-\beta=1}$

$x_{6}\neq 0$$x_{5}=0$$\stackrel{{\scriptstyle\alpha=\beta}}{{x_{5}\neq
0}}$$x_{3}=0$$\stackrel{{\scriptstyle\beta=1}}{{x_{3}\neq
0}}$$x_{3}=0$$\stackrel{{\scriptstyle\alpha=1}}{{x_{3}\neq
0}}$$x_{1}=0$$\stackrel{{\scriptstyle\alpha=-1\,{\rm
or}\,\alpha+\beta=-1}}{{x_{1}\neq
0}}$$x_{3}x_{4}+x_{1}x_{6}=0$$\stackrel{{\scriptstyle{\rm
Isomorphic\,to\,XIII}_{\alpha=-\beta=1}}}{{\alpha=-1,x_{3}x_{4}+x_{1}x_{6}\neq
0}}$$x_{1}x_{6}-x_{2}x_{5}=0$$\stackrel{{\scriptstyle\alpha=\beta=-1/2}}{{x_{1}x_{6}-x_{2}x_{5}\neq
0}}$$x_{1}x_{6}-x_{2}x_{5}+x_{3}x_{4}=0$$\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{x_{1}x_{6}-x_{2}x_{5}+x_{3}x_{4}\neq 0}}$XIV${\rm
XV}^{\alpha=-1}_{\alpha+\beta=-1}$${\rm XVI}_{\beta=1}$${\rm
XVII}_{\alpha=\beta}$${\rm XVIII}_{\alpha=\beta=-1/2}$${\rm
XIX}_{\alpha=\beta=1}$

We now study four subcases: a) $\alpha=\beta=1$, b) $\alpha=1\neq\beta$, c)
$\alpha=\beta\neq 1$, d) remaining non-isomorphic cases.

Case d): In this case, ${\rm ad}_{e_{4}}$ acts on $\langle
e_{1},e_{2},e_{3}\rangle$ having three different eigenvalues and this leads to
the fact that the only derivations are those common to all $\alpha,\beta$,
i.e. $\mathfrak{der}(\mathfrak{s}^{\alpha,\beta}_{3})=\mathfrak{der}_{\rm
all}(\mathfrak{s}_{3})$. The connected parts of the loci of the Darboux
families of the above Darboux tree can be found in Table 2. Such connected
parts are the orbits of ${\rm Aut}_{c}(\mathfrak{s}^{\alpha,\beta}_{3})$ in
$\mathcal{Y}_{\mathfrak{s}^{\alpha,\beta}_{3}}$ for $\alpha,\beta,1$ taking
different values. Let us obtain the classification of $r$-matrices up to
elements of ${\rm Aut}(\mathfrak{s}^{\alpha,\beta}_{3})$. On the one hand, the
Lie algebra automorphism group reads

${\rm
Aut}(\mathfrak{s}^{\alpha,\beta}_{3})=\left\\{\left(\begin{array}[]{cccc}T^{1}_{1}&0&0&T^{4}_{1}\\\
0&T^{2}_{2}&0&T^{4}_{2}\\\ 0&0&T^{3}_{3}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T^{1}_{1},T^{2}_{2},T^{3}_{3}\in\mathbb{R}\backslash\\{0\\},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\right\\}.$

As in previous sections, we only need for our purposes one element of each
connected part of ${\rm Aut}(\mathfrak{s}^{\alpha,\beta}_{3})$, which has
eight ones. An element of each connected component and its lift to
$\Lambda^{2}\mathfrak{s}_{3}$ are given by

$T_{\lambda_{1},\lambda_{2},\lambda_{3}}:=\left(\begin{array}[]{cccc}\lambda_{1}&0&0&0\\\
0&\lambda_{2}&0&0\\\ 0&0&\lambda_{3}&0\\\
0&0&0&1\end{array}\right)\Rightarrow\Lambda^{2}T_{\lambda_{1},\lambda_{2},\lambda_{3}}:={\small\left(\begin{array}[]{cccccc}\lambda_{1}\lambda_{2}&0&0&0&0&0\\\
0&\lambda_{1}\lambda_{3}&0&0&0&0\\\ 0&0&\lambda_{1}&0&0&0\\\
0&0&0&\lambda_{2}\lambda_{3}&0&0\\\ 0&0&0&0&\lambda_{2}&0\\\
0&0&0&0&0&\lambda_{3}\\\
\end{array}\right)},\quad\lambda_{1},\lambda_{2},\lambda_{3}\in\\{\pm 1\\}.$
(7.3)

By using (7.3), we can verify whether some of the strata of
$\mathscr{E}_{\mathfrak{s}_{3}}$ in
$\mathcal{Y}_{\mathfrak{s}_{3}^{\alpha,\beta}}$ are still connected by a Lie
algebra automorphism of $\mathfrak{s}^{\alpha,\beta}_{3}$. The results are
summarised in Table 2. To obtain the equivalence classes of coboundary
coproducts for each $\alpha,\beta$, it is enough to identify the orbits in
Table 2 whose elements are the same up to an element of
$(\Lambda^{2}\mathfrak{s}^{\alpha,\beta}_{3})^{\mathfrak{s}^{\alpha,\beta}_{3}}$.
In particular, we have the following subcases:

  * •

Case $\alpha=-1,\beta\neq 1$. Hence,
$(\Lambda^{2}\mathfrak{s}^{-1,\beta}_{3})^{\mathfrak{s}^{-1,\beta}_{3}}=\langle
e_{12}\rangle$. By analysing Table 2, we obtain the coboundary coproduct
classes

$a)\,{\rm I},\,b)\,{\rm II},{\rm III},\,c)\,{\rm IV},{\rm V},\,d)\,{\rm
VI},{\rm VII}_{\pm},\,e)\,{\rm VIII},\,f)\,{\rm X},\,g)\,{\rm XIV},{\rm
XV}_{\alpha=-1}.$

  * •

Case $\alpha\neq-1$ and $\beta\neq-1$. Since
$(\Lambda^{2}\mathfrak{s}^{\alpha,\beta}_{3})^{\mathfrak{s}^{\alpha,\beta}_{3}}=0$,
the classes of equivalent coboundary coproducts are given by the orbits of
${\rm Aut}(\mathfrak{s}^{\alpha,\beta}_{3})$ within
$\mathcal{Y}_{\mathfrak{s}^{\alpha,\beta}_{3}}$. In this case, we have the
classes of equivalent coboundary coproducts given by

${\rm I},{\rm II},{\rm III},{\rm IV},{\rm V},{\rm VI},{\rm VII}_{+},{\rm
VII}_{-},{\rm VIII},{\rm IX}_{\alpha+\beta\in\\{0,-1\\}},{\rm X},{\rm
XI}_{\alpha+\beta=-1},{\rm XIV},{\rm XV}_{\alpha+\beta=-1}.$

Case c): This time $\alpha=\beta\neq 1$. Then, we have

${\rm
Aut}(\mathfrak{s}^{\alpha,\alpha}_{3})=\left\\{\left(\begin{array}[]{cccc}T^{1}_{1}&0&0&T^{4}_{1}\\\
0&T^{2}_{2}&T_{2}^{3}&T^{4}_{2}\\\ 0&T_{3}^{2}&T^{3}_{3}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T^{1}_{1},T^{2}_{2}T^{3}_{3}-T^{2}_{3}T^{3}_{2}\in\mathbb{R}\backslash\\{0\\},T_{1}^{1},T^{2}_{2},T^{3}_{3},T_{2}^{3},T_{3}^{2},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\right\\}.$

To obtain the orbits of ${\rm Aut}(\mathfrak{s}^{\alpha,\alpha}_{3})$ on
$\mathcal{Y}_{\mathfrak{s}^{\alpha,\alpha}_{3}}$ from the orbits of ${\rm
Aut}_{{\rm all},c}(\mathfrak{s}_{3})$, it is necessary to write ${\rm
Aut}(\mathfrak{s}^{\alpha,\alpha}_{3})$ as a composition of ${\rm Aut}_{{\rm
all},c}(\mathfrak{s}_{3})$ with certain Lie algebra automorphisms of
$\mathfrak{s}^{\alpha,\alpha}_{3}$ so that their composition generates ${\rm
Aut}(\mathfrak{s}^{\alpha,\alpha}_{3})$. This can be done by using the Lie
algebra automorphisms of $\mathfrak{s}_{3}^{\alpha,\alpha}$ of the form
$T_{A}:={\rm Id}\otimes A{\otimes}{\rm Id}$, for $A\in GL(2,\mathbb{R})$.
Then, $\Lambda^{2}T_{A}=A\otimes{\rm Id}\otimes(\det A){\rm Id}\otimes A$. By
taking the action of these $\Lambda^{2}T_{A}$ on the strata of the
distribution spanned by $V^{\rm all}_{\mathfrak{s}_{3}}$ in
$\mathcal{Y}_{\mathfrak{s}^{\alpha,\alpha}_{3}}$, we obtain the orbits of
${\rm Aut}(\mathfrak{s}^{\alpha,\alpha}_{3})$ on
$\mathcal{Y}_{\mathfrak{s}^{\alpha,\alpha}_{3}}$. Our results are summarised
in Table 2.

To obtain the equivalence classes of coboundary coproducts for each
$\alpha=\beta$, it is enough to identify the orbits in Table 2 whose elements
are the same up to an element of
$(\Lambda^{2}\mathfrak{s}^{\alpha,\alpha}_{3})^{\mathfrak{s}^{\alpha,\alpha}_{3}}=0$.
Therefore, the result is given by the classes of equivalent $r$-matrices
detailed in Table 2.

Case b): This time $\alpha=1\neq\beta$. Therefore,

${\rm
Aut}(\mathfrak{s}^{1,\beta}_{3})=\left\\{\left(\begin{array}[]{cccc}T^{1}_{1}&T_{1}^{2}&0&T^{4}_{1}\\\
T_{2}^{1}&T^{2}_{2}&0&T^{4}_{2}\\\ 0&0&T^{3}_{3}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T^{3}_{3},T^{1}_{1}T^{2}_{2}-T^{1}_{2}T^{2}_{1}\in\mathbb{R}\backslash\\{0\\},T_{1}^{1},T_{2}^{2},T_{1}^{2},T_{2}^{1},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\right\\}.$

To derive the orbits of ${\rm Aut}(\mathfrak{s}^{1,\beta}_{3})$ on
$\mathcal{Y}_{\mathfrak{s}^{1,\beta}_{3}}$ from the orbits of ${\rm Aut}_{{\rm
all},c}(\mathfrak{s}_{3})$, we again write ${\rm
Aut}(\mathfrak{s}^{1,\beta}_{3})$ as a composition of ${\rm Aut}_{{\rm
all},c}(\mathfrak{s}_{3})$ with certain Lie algebra automorphisms of
$\mathfrak{s}^{1,\beta}_{3}$ so that their composition generates ${\rm
Aut}(\mathfrak{s}^{1,\beta}_{3})$. This can be done by employing the Lie
algebra autmorphisms of $\mathfrak{s}^{1,\beta}_{3}$ given by
$T_{A}:=A{\otimes}{\rm Id}\otimes{\rm Id}$, for $A\in GL(2,\mathbb{R})$. Then
$\Lambda^{2}T_{A}=(\det A){\rm Id}\otimes(\tau_{43}\circ A\otimes
A\circ\tau_{43})\otimes{\rm Id}$, where $\tau_{43}$ is the permutation of
coordinates three and four in $\Lambda^{2}\mathfrak{s}^{1,\beta}_{3}$. By
taking the action of these $\Lambda^{2}T_{A}$ on the strata of the
distribution spanned by $V^{\rm all}_{\mathfrak{s}_{3}}$ in
$\mathcal{Y}_{\mathfrak{s}^{1,\beta}_{3}}$, we obtain the orbits of ${\rm
Aut}(\mathfrak{s}^{1,\beta}_{3})$ on
$\mathcal{Y}_{\mathfrak{s}^{1,\beta}_{3}}$. Our results are summarised in
Table 2.

To obtain the equivalence classes of coboundary coproducts for each $\beta$,
it is enough to identify the equivalence classes of $r$-matrices in Table 2
whose elements are the same up to an element of
$(\Lambda^{2}\mathfrak{s}^{1,\beta}_{3})^{\mathfrak{s}^{1,\beta}_{3}}$. If
$\beta\neq-1$, these are just the induced by the orbits of ${\rm
Aut}(\mathfrak{s}^{1,\beta}_{3})$ on
$\mathcal{Y}_{\mathfrak{s}^{1,\beta}_{3}}$. Otherwise,
$(\Lambda^{2}\mathfrak{s}^{1,-1}_{3})^{\mathfrak{s}^{1,-1}_{3}}=\langle
e_{13},e_{23}\rangle$ and

$a)\,\mathscr{I},\mathscr{III},\,b)\mathscr{II}\,{\rm(zero-
class)},\,c)\,\mathscr{IV},\,d)\,\mathscr{V}_{\beta=-1}\,e)\,\mathscr{VI}.$

Case a): In this case, it is easier to derive the Darboux families for the Lie
algebra of derivations.

$\mathfrak{der}_{\rm
all}(\mathfrak{s}_{3})=\left\\{\left(\begin{array}[]{cccc}A&v\\\
0&0\end{array}\right):A\in\mathfrak{gl}(3,\mathbb{R}),v\in\mathbb{R}^{3}\right\\}.$

Due to the larger family of symmetries for $\alpha=\beta=1$, the Lie algebra
$V_{\mathfrak{s}_{3}^{1,1}}$ is spanned by

$\begin{gathered}X_{1}=x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}},\quad
X_{2}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\quad
X_{3}=x_{1}\partial_{x_{1}}+x_{4}\partial_{x_{4}}+x_{5}\partial_{x_{5}},\\\
X_{4}=x_{3}\partial_{x_{1}}-x_{6}\partial_{x_{4}},\quad
X_{5}=x_{2}\partial_{x_{2}}+x_{4}\partial_{x_{4}}+x_{6}\partial_{x_{6}},\quad
X_{6}=x_{3}\partial_{x_{2}}+x_{5}\partial_{x_{4}},\\\
X_{7}=x_{4}\partial_{x_{2}}+x_{5}\partial_{x_{3}},\quad
X_{8}=-x_{4}\partial_{x_{1}}+x_{6}\partial_{x_{3}},\quad
X_{9}=x_{2}\partial_{x_{4}}+x_{3}\partial_{x_{5}},\quad
X_{10}=x_{2}\partial_{x_{1}}+x_{6}\partial_{x_{5}},\\\
X_{11}=-x_{1}\partial_{x_{4}}+x_{3}\partial_{x_{6}},\quad
X_{12}=x_{1}\partial_{x_{2}}+x_{5}\partial_{x_{6}}.\end{gathered}$

The Darboux tree is very simple and becomes

$x_{3}x_{4}-x_{2}x_{5}+x_{1}x_{6}=0$$\stackrel{{\scriptstyle{\rm
No\,solution}}}{{x_{3}x_{4}-x_{2}x_{5}+x_{1}x_{6}\neq
0}}$$x_{3}=0,x_{5}=0,x_{6}=0$$x_{3}^{2}+x_{5}^{2}+x_{6}^{2}\neq
0$$x_{2}=0,x_{1}=0,x_{4}=0$$x_{2}^{2}+x_{1}^{2}+x_{4}^{2}\neq 0$0iii

It is immediate that the above gives rise to two non-zero coboundaries given
in Table 2. Since in this case
$(\Lambda^{2}\mathfrak{s}^{1,1}_{3})^{\mathfrak{s}^{1,1}_{3}}=0$, each class
of equivalent $r$-matrices amounts to a class of equivalent coboundary
coproducts.

###  7.4 Lie algebra $\mathfrak{s}_{4}$

Tables 1, 3–5 contain the necessary information on the structure constants of
$\mathfrak{s}_{4}$ and several Schouten brackets to prove our following
results. Recall that $\alpha\in\mathbb{R}\backslash\\{0\\}$. First,
$(\Lambda^{3}\mathfrak{s}_{4})^{\mathfrak{s}_{4}}=\langle e_{123}\rangle$ for
$2+\alpha=0$ and it is zero for remaining allowed values of $\alpha$.
Meanwhile, $(\Lambda^{2}\mathfrak{s}_{4})^{\mathfrak{s}_{4}}=0$ for
$\alpha\neq-1$ and $(\Lambda^{2}\mathfrak{s}_{4})^{\mathfrak{s}_{4}}=\langle
e_{13}\rangle$ for $\alpha=-1$.

As the Lie algebras of the class $\mathfrak{s}_{4}$ depend on a parameter
$\alpha\in\mathbb{R}\backslash\\{0\\}$, the space of derivations depend on
$\alpha$. It is indeed the same for all values of
$\alpha\in\mathbb{R}\backslash\\{0,1\\}$, and it becomes larger for
$\alpha=1$. Due to this, We shall proceed as in Subsection 7.3. By Remark 2.1,
the space of derivations of $\mathfrak{s}_{4}$ for all possible values of
$\alpha$ read

$\mathfrak{der}_{\rm
all}(\mathfrak{s}_{4}):=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&0&\mu_{14}\\\
0&\mu_{11}&0&\mu_{24}\\\ 0&0&\mu_{33}&\mu_{34}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{33},\mu_{14},\mu_{24},\mu_{34}\in\mathbb{R}\right\\}.$

By extending the previous derivations to $\Lambda^{2}\mathfrak{s}_{4}$ and
using Remark 2.1, we obtain a basis of $V^{\rm all}_{\mathfrak{s}_{4}}$ of the
form

$\begin{gathered}X_{1}=2x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}}+x_{4}\partial_{x_{4}}+x_{5}\partial_{x_{5}},\quad
X_{2}=x_{4}\partial_{x_{2}}+x_{5}\partial_{x_{3}},\quad
X_{3}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\\\
X_{4}=x_{3}\partial_{x_{1}}-x_{6}\partial_{x_{4}},\quad
X_{5}=x_{2}\partial_{x_{2}}+x_{4}\partial_{x_{4}}+x_{6}\partial_{x_{6}},\quad
X_{6}=x_{3}\partial_{x_{2}}+x_{5}\partial_{x_{4}}.\end{gathered}$

We recall that these vector fields span the Lie algebra of fundamental vector
fields of the action on $\Lambda^{2}\mathfrak{s}_{4}$ of the Lie algebra
automorphisms ${\rm Aut}_{\rm all}(\mathfrak{s}_{4})$ that are common for all
values of $\alpha\in\mathbb{R}\backslash\\{0\\}$.

For an element $r\in\Lambda^{2}\mathfrak{s}_{4}$, we get

$[r,r]=2[2x_{1}x_{6}-(1+\alpha)x_{2}x_{5}+(1+\alpha)x_{3}x_{4}-x_{4}x_{5}]e_{123}-2x_{5}^{2}e_{124}+2[-(1-\alpha)x_{3}x_{6}-x_{5}x_{6}]e_{134}+2(\alpha-1)x_{5}x_{6}e_{234}.$

If $\alpha\neq-2$, we have
$(\Lambda^{3}\mathfrak{s}_{4})^{\mathfrak{s}_{4}}=0$. Thus, the mCYBE and the
CYBE are equal in this case and they read

$2x_{1}x_{6}+(1+\alpha)x_{3}x_{4}=0,\quad(1-\alpha)x_{3}x_{6}=0,\quad
x_{5}=0.$

For the case $\alpha+2=0$, we have
$(\Lambda^{3}\mathfrak{s}_{4})^{\mathfrak{s}_{4}}=\langle e_{123}\rangle$.
Thus, the mCYBE reads

$\quad(1-\alpha)x_{3}x_{6}=0,\quad x_{5}=0.$

Since $x_{5},x_{6}$ are the bricks for $\mathfrak{s}_{4}$, our Darboux tree
for the Darboux families starts with the cases $x_{i}=0$ and $x_{i}\neq 0$,
$i\in\\{5,6\\}$. The full Darboux tree is presented below.

$x_{6}=0$$x_{5}=0$$\stackrel{{\scriptstyle{\rm No\,solutions}}}{{x_{5}\neq
0}}$$x_{4}=0$$x_{4}\neq 0$$x_{3}=0$$x_{3}\neq
0$$x_{3}=0$$\stackrel{{\scriptstyle\alpha=-2\lor\alpha=-1}}{{x_{3}\neq
0}}$$x_{2}=0$$x_{2}\neq 0$$x_{1}=0$$x_{1}\neq 0$$x_{1}=0$$x_{1}\neq
0$$x_{1}=0$$x_{1}\neq 0$0IIIIIIIVVVI${\rm
VII}_{\alpha\in\\{-2,-1\\}}$$x_{6}\neq 0$$x_{5}=0$$\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{x_{5}\neq
0}}$$x_{3}=0$$\stackrel{{\scriptstyle\alpha=1}}{{x_{3}\neq
0}}$$x_{1}=0$$\stackrel{{\scriptstyle\alpha=-2}}{{x_{1}\neq
0}}$$x_{3}x_{4}+x_{1}x_{6}=0$$\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{x_{3}x_{4}+x_{1}x_{6}\neq 0}}$VIII${\rm
IX}_{\alpha=-2}$${\rm X}_{\alpha=1}$

The connected parts of the subspaces denoted in the above diagram are the
orbits of ${\rm Aut}_{\rm all,c}(\mathfrak{s}_{4})$ on
$\mathcal{Y}_{\mathfrak{s}_{4}}$. To obtain the orbits of ${\rm Aut}_{\rm
all}(\mathfrak{s}_{4})$ on $\mathcal{Y}_{\mathfrak{s}_{4}}$, where

${\rm Aut}_{\rm
all}(\mathfrak{s}_{4})=\left\\{\left(\begin{array}[]{cccc}T^{2}_{2}&T^{2}_{1}&0&T^{4}_{1}\\\
0&T^{2}_{2}&0&T^{4}_{2}\\\ 0&0&T^{3}_{3}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T^{2}_{2},T^{3}_{3}\in\mathbb{R}\backslash\\{0\\},\,T^{2}_{1},T_{1}^{4},T_{2}^{4},T_{3}^{4}\in\mathbb{R}\right\\},$

we verify whether some of the connected components of the orbits of ${\rm
Aut}_{\rm all,c}(\mathfrak{s}_{4})$ on $\mathcal{Y}_{\mathfrak{s}_{4}}$ are
additionally connected by an element of ${\rm Aut}_{\rm
all}(\mathfrak{s}_{4})$. To do so, we use, as previously, the lift to
$\Lambda^{2}\mathfrak{s}_{4}$ of one element of each connected component of
${\rm Aut}_{\rm all}(\mathfrak{s}_{4})$, for instance

$T_{\lambda_{1},\lambda_{2}}:=\left(\begin{array}[]{cccc}\lambda_{1}&0&0&0\\\
0&\lambda_{1}&0&0\\\ 0&0&\lambda_{2}&0\\\
0&0&0&1\end{array}\right)\Rightarrow\Lambda^{2}T_{\lambda_{1},\lambda_{2}}={\small\left(\begin{array}[]{cccccc}1&0&0&0&0&0\\\
0&\lambda_{1}\lambda_{2}&0&0&0&0\\\ 0&0&\lambda_{1}&0&0&0\\\
0&0&0&\lambda_{1}\lambda_{2}&0&0\\\ 0&0&0&0&\lambda_{1}&0\\\
0&0&0&0&0&\lambda_{2}\\\
\end{array}\right)},\quad\lambda_{1},\lambda_{2}\in\\{\pm 1\\}.$ (7.4)

For $\alpha\in\mathbb{R}\backslash\\{1,0\\}$, one has that ${\rm
Aut}(\mathfrak{s}^{\alpha}_{4})={\rm Aut}_{\rm all}(\mathfrak{s}_{4})$, where
$\mathfrak{s}^{\alpha}_{4}$ stands for the Lie algebra $\mathfrak{s}_{4}$ for
a fixed value of $\alpha$. Then, the classes of equivalent $r$-matrices (up to
Lie algebra automorphisms of $\mathfrak{s}_{4}^{1}$) on the Lie algebra
$\mathfrak{s}^{\alpha}_{4}$ can easily be obtained and they are summarised in
Table 2. Moreover, the classes of equivalent coboundary coproducts for
$\alpha=-1$ are induced by the families

$a)\,{\rm II}({\rm zero\,class}),\,b)\,{\rm I}_{+},{\rm III}_{+},\,c)\,{\rm
I}_{-},{\rm III}_{-},\,d)\,{\rm IV},\,e)\,{\rm VII}_{\alpha=-1},\,f)\,{\rm
V},\,g)\,{\rm VI}_{+},\,h)\,{\rm VI}_{-},\,i)\,{\rm VIII}.$

For those $\mathfrak{s}_{4}^{\alpha}$ with $|\alpha|\neq 1$, one has
$(\Lambda^{2}\mathfrak{s}_{4}^{\alpha})^{\mathfrak{s}_{4}^{\alpha}}=0$ and the
classes of equivalent coboundary coproducts are given by each one of the
following subsets

$a)\,{\rm I}_{-},\,b)\,{\rm I}_{+},\,c)\,{\rm II},\,d)\,{\rm
III}_{+},\,e)\,{\rm III}_{-},\,f)\,{\rm IV},\,g)\,{\rm V},\,h)\,{\rm
VI}_{+},\,i)\,{\rm VI}_{-},\,j)\,{\rm VII}_{\alpha=-2},\,k)\,{\rm
VIII},\,l)\,{\rm IX}_{-}^{\alpha=-2},\,m)\,{\rm IX}_{+}^{\alpha=-2}.$

Note that, for values $|\alpha|\neq 1$, not all the above classes may be
simultaneously available as they arise for particular values of $\alpha$.

Let us consider now the case of the Lie algebra $\mathfrak{s}^{1}_{4}$, i.e.
the Lie algebra of the class $\mathfrak{s}_{4}$ for $\alpha=1$. In this case,
the group of Lie algebra automorphisms is slightly larger than for remaining
admissible values of $\alpha$. In particular,

${\rm
Aut}(\mathfrak{s}^{1}_{4})=\left\\{\left(\begin{array}[]{cccc}T^{2}_{2}&T^{2}_{1}&T^{3}_{1}&T^{4}_{1}\\\
0&T^{2}_{2}&0&T^{4}_{2}\\\ 0&T_{3}^{2}&T^{3}_{3}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T^{2}_{2},T^{3}_{3}\in\mathbb{R}\backslash\\{0\\},\,T_{3}^{2},T^{2}_{1},T^{3}_{1},T_{1}^{4},T_{2}^{4},T_{3}^{4}\in\mathbb{R}\right\\}$

and ${\rm Aut}(\mathfrak{s}^{1}_{4})$ is the group resulting of composing
${\rm Aut}_{\rm all}(\mathfrak{s}_{4})$ with the Lie algebra automorphisms of
$\mathfrak{s}_{4}^{1}$ of the form

$T(e_{1})=e_{1},\,T(e_{2})=e_{2}+\lambda e_{3},\,T(e_{3})=e_{3}+\mu
e_{1},\,T(e_{4})=e_{4},\qquad\forall\lambda,\mu\in\mathbb{R}.$

Hence, to obtain the orbits of the action ${\rm Aut}(\mathfrak{s}_{4})$ on
$\mathcal{Y}_{\mathfrak{s}_{4}}$, it is enough to act $\Lambda^{2}T_{A}$ on
the orbits of ${\rm Aut}_{\rm all}(\mathfrak{s}_{4})$. Note that

$\begin{gathered}\Lambda^{2}T(e_{12})=e_{12}+\lambda
e_{13},\,\,\Lambda^{2}T(e_{13})=e_{13},\,\Lambda^{2}T(e_{14})=e_{14},\,\\\
\Lambda^{2}T(e_{23})=e_{23}-e_{12}\mu-\lambda\mu
e_{13},\,\Lambda^{2}T(e_{24})=e_{24}+\lambda
e_{34},\,\Lambda^{2}T_{34}=e_{34}+\mu e_{14},\end{gathered}$

for every $\lambda,\mu\in\mathbb{R}$. This will give the final orbits detailed
in Table 2. Since
$(\Lambda^{2}\mathfrak{s}^{1}_{4})^{\mathfrak{s}^{1}_{4}}=0$, the families of
equivalent coboundary coproducts are given by the ones induced by the families
of equivalent $r$-matrices.

###  7.5 Lie algebra $\mathfrak{s}_{5}$

Structure constants for Lie algebra $\mathfrak{s}_{5}$ are given in Table 1.
Using this information, one can compute the Schouten brackets between the
elements of $\mathfrak{s}_{5}$, $\Lambda^{2}\mathfrak{s}_{5}$, and
$\Lambda^{3}\mathfrak{s}_{5}$ from the information contained in Tables 3–5. In
particular, we obtain
$(\Lambda^{2}\mathfrak{s}_{5})^{\mathfrak{s}_{5}}=\langle e_{23}\rangle$ for
$\beta=0$ and $(\Lambda^{2}\mathfrak{s}_{5})^{\mathfrak{s}_{5}}=\\{0\\}$ for
$\beta\neq 0$. Moreover, $(\Lambda^{3}\mathfrak{s}_{5})^{\mathfrak{s}_{5}}=0$
for $\alpha+2\beta\neq 0$ and
$(\Lambda^{3}\mathfrak{s}_{5})^{\mathfrak{s}_{5}}=\langle e_{123}\rangle$
otherwise.

By Remark 2.1, we obtain that the derivations of $\mathfrak{s}_{5}$ read

$\mathfrak{der}(\mathfrak{s}_{5}):=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&0&0&\mu_{14}\\\
0&\mu_{22}&\mu_{23}&\mu_{24}\\\ 0&-\mu_{23}&\mu_{22}&\mu_{34}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{22},\mu_{23},\mu_{14},\mu_{24},\mu_{34}\in\mathbb{R}\right\\},$

which give rise to the basis of $V_{\mathfrak{s}_{5}}$ of the form

$\begin{gathered}X_{1}=x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}},\quad
X_{2}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\quad
X_{3}=x_{1}\partial_{x_{1}}+x_{2}\partial_{x_{2}}+2x_{4}\partial_{x_{4}}+x_{5}\partial_{x_{5}}+x_{6}\partial_{x_{6}},\\\
X_{4}=x_{2}\partial_{x_{1}}-x_{1}\partial_{x_{2}}+x_{6}\partial_{x_{5}}-x_{5}\partial_{x_{6}},\quad
X_{5}=x_{3}\partial_{x_{1}}-x_{6}\partial_{x_{4}},\quad
X_{6}=x_{3}\partial_{x_{2}}+x_{5}\partial_{x_{4}}.\end{gathered}$

For an element $r\in\Lambda^{2}\mathfrak{s}_{5}$, we get

$\begin{split}[r,r]&=2[x_{1}x_{5}+(\alpha+\beta)x_{1}x_{6}-(\alpha+\beta)x_{2}x_{5}+x_{2}x_{6}+2\beta
x_{3}x_{4}]e_{123}+2[(\beta-\alpha)x_{3}x_{5}+x_{3}x_{6}]e_{124}\\\
&+2[-x_{3}x_{5}+(\beta-\alpha)x_{3}x_{6}]e_{134}-2(x_{5}^{2}+x_{6}^{2})e_{234}.\end{split}$

If $\alpha+2\beta\neq 0$, then
$(\Lambda^{3}\mathfrak{s}_{5})^{\mathfrak{s}_{5}}=0$. Thus, the mCYBE and the
CYBE are equal in this case and they read

$\beta x_{3}x_{4}=0,\quad x_{5}=0,\quad x_{6}=0.$

For $\alpha+2\beta=0$, we have
$(\Lambda^{3}\mathfrak{s}_{5})^{\mathfrak{s}_{5}}=\langle e_{123}\rangle$ and
the mCYBE reads

$x_{5}=0,\quad x_{6}=0.$

Since $x_{3}$ is the only brick for $\mathfrak{s}_{5}$, we start our Darboux
tree with the cases $x_{3}=0$ and $x_{3}\neq 0$. The full Darboux tree is
presented below.

$x_{3}=0$$x_{6}=0$$\stackrel{{\scriptstyle{\rm No\,solutions}}}{{x_{6}\neq
0}}$$x_{5}=0$$\stackrel{{\scriptstyle{\rm No\,solutions}}}{{x_{5}\neq
0}}$$x_{4}=0$$x_{4}\neq 0$$x_{1}^{2}+x_{2}^{2}=0$$x_{1}^{2}+x_{2}^{2}\neq
0$$x_{1}^{2}+x_{2}^{2}=0$$x_{1}^{2}+x_{2}^{2}\neq 0$0IIIIII$x_{3}\neq
0$$x_{6}=0$$\stackrel{{\scriptstyle{\rm No\,solutions}}}{{x_{6}\neq
0}}$$x_{5}=0$$\stackrel{{\scriptstyle{\rm No\,solutions}}}{{x_{5}\neq
0}}$$x_{4}=0$$\stackrel{{\scriptstyle\beta=0\lor\alpha+2\beta=0}}{{x_{4}\neq
0}}$IV${\rm V}^{\alpha=-2\beta}_{\beta=0}$

Orbits of ${\rm Aut}_{c}(\mathfrak{s}_{5})$ are given by the connected parts
of the loci of the above Darboux tree. Solutions are described in Table 2.

The automorphism group of $\mathfrak{s}_{5}$ reads

${\rm
Aut}(\mathfrak{s}_{5})=\left\\{\left(\begin{array}[]{cccc}T^{1}_{1}&0&0&T^{4}_{1}\\\
0&T^{2}_{2}&T^{3}_{2}&T^{4}_{2}\\\ 0&-T^{3}_{2}&T^{2}_{2}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):T^{1}_{1}\in\mathbb{R},(T^{2}_{2})^{2}+(T^{3}_{2})^{2}>0,\,T_{2}^{2},T_{2}^{3},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\right\\}.$

Since the subgroup of $GL(2,\mathbb{R})$ of matrices of the form

$\left(\begin{array}[]{cc}T^{2}_{2}&T^{3}_{2}\\\
-T^{3}_{2}&T^{2}_{2}\end{array}\right)\in
GL(2,\mathbb{R}),\qquad(T^{2}_{2})^{2}+(T^{3}_{2})^{2}>0,$

can be parametrised via $\phi\in[0,2\pi[$ and
$\mu:=[(T^{2}_{2})^{2}+(T^{3}_{2})^{2}]^{1/2}\in\mathbb{R}_{+}$ by setting
$T^{2}_{2}=\mu\cos\phi$ and $T^{3}_{2}=\mu\sin\phi$, there are two connected
components of ${\rm Aut}(\mathfrak{s}_{5})$. A representative of each
connected part and its lift to $\Lambda^{2}\mathfrak{s}_{5}$ read

$T_{\lambda}:=\left(\begin{array}[]{cccc}\lambda&0&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\
0&0&0&1\end{array}\right)\Longrightarrow\Lambda^{2}T_{\lambda}:=\left(\begin{array}[]{cccccc}\lambda&0&0&0&0&0\\\
0&\lambda&0&0&0&0\\\ 0&0&\lambda&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\
0&0&0&0&0&1\\\ \end{array}\right),\quad\lambda\in\\{\pm 1\\}.$ (7.5)

Using techniques from previous sections, we can easily verify whether the
orbits of ${\rm Aut}_{c}(\mathfrak{s}_{5})$ are additionally connected by a
Lie algebra automorphism of $\mathfrak{s}_{5}$ via $\Lambda^{2}T_{\lambda}$.
Our results are summarised in Table 2. Moreover, for $\beta\neq 0$, the
classes of coboundary coproducts are induced from the following families of
$r$-matrices:

${a)}\,{\rm I},\,{b)}\,{\rm II}_{+},{c)}\,{\rm II}_{-},\,{d)}\,{\rm
III}_{+},\,{e)}\,{\rm III}_{-},\,{f)}\,{\rm IV},\,{g)}\,{\rm
V}^{\alpha=-2\beta}_{+},\,{h)}\,{\rm V}^{\alpha=-2\beta}_{-}.$

Meanwhile, for $\beta=0$, the list of families of equivalents coboundary
coproducts read

${a)}\,{\rm II}_{\pm},\,{b)}\,{\rm I},{\rm III}_{\pm}\,,c)\,{\rm IV},{\rm
V}_{\pm}^{\beta=0}.$

###  7.6 Lie algebra $\mathfrak{s}_{6}$

Structure constants of Lie algebra $\mathfrak{s}_{6}$ are given in Table 1.
Using this information, we can compute the Schouten brackets of elements of
$\mathfrak{s}_{6}$, $\Lambda^{2}\mathfrak{s}_{6}$, and
$\Lambda^{3}\mathfrak{s}_{6}$ (see Tables 3–5). As in previous sections,
Tables 1,3–5 contain the necessary information to accomplish the following
calculations. In particular, we have
$(\Lambda^{3}\mathfrak{s}_{6})^{\mathfrak{s}_{6}}=\langle e_{123}\rangle$
while $(\Lambda^{2}\mathfrak{s}_{6})^{\mathfrak{s}_{6}}=\\{0\\}$.

By Remark 2.1, one obtains that

$\mathfrak{der}(\mathfrak{s}_{6}):=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&\mu_{13}&\mu_{14}\\\
0&\mu_{22}&0&-\mu_{13}\\\ 0&0&\mu_{11}-\mu_{22}&-\mu_{12}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{13},\mu_{14},\mu_{22}\in\mathbb{R}\right\\},$

which gives rise to the basis of $V_{\mathfrak{s}_{6}}$ of the form

$\begin{gathered}X_{1}=x_{1}\partial_{x_{1}}+2x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}}+x_{4}\partial_{x_{4}}+x_{6}\partial_{x_{6}},\quad
X_{2}=(-x_{3}+x_{4})\partial_{x_{2}}+x_{5}\partial_{x_{3}}-x_{5}\partial_{x_{4}},\\\
X_{3}=(-x_{3}-x_{4})\partial_{x_{1}}+x_{6}\partial_{x_{3}}+x_{6}\partial_{x_{4}},\quad
X_{4}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\quad
X_{5}=x_{1}\partial_{x_{1}}-x_{2}\partial_{x_{2}}+x_{5}\partial_{x_{5}}-x_{6}\partial_{x_{6}}.\end{gathered}$

For an element $r\in\Lambda^{2}\mathfrak{s}_{6}$, we get

$[r,r]=2(x_{1}x_{6}+x_{2}x_{5}+x_{4}^{2})e_{123}+2(x_{3}+x_{4})x_{5}e_{124}+2(x_{4}-x_{3})x_{6}e_{134}-4x_{5}x_{6}e_{234}.$

Since $(\Lambda^{3}\mathfrak{s}_{6})^{\mathfrak{s}_{6}}=\langle
e_{123}\rangle$, the mCYBE reads

$(x_{3}+x_{4})x_{5}=0\quad(x_{4}-x_{3})x_{6}=0,\quad x_{5}x_{6}=0,$

whereas the CYBE is

$x_{1}x_{6}+x_{2}x_{5}+x_{4}^{2}=0,\quad(x_{3}+x_{4})x_{5}=0,\quad(x_{4}-x_{3})x_{6}=0,\quad
x_{5}x_{6}=0.$

Note that the CYBE obtained above is exactly the result obtained in [6, eq.
(3.7)] under the substitution $x_{5}=-\alpha_{+}$, $x_{6}=-\alpha_{-}$,
$x_{4}=\xi$ and $x_{3}=\vartheta$.

Since $x_{5},x_{6}$ are the bricks of $\mathfrak{s}_{6}$, our Darboux tree
starts with cases $x_{i}=0$ and $x_{i}\neq 0$, for $i\in\\{5,6\\}$. The full
Darboux tree is presented below.

$x_{6}=0$$x_{5}=0$$x_{5}\neq 0$$x_{4}=0$$x_{4}\neq
0$$\stackrel{{\scriptstyle\tiny{\rm No\,solutions}}}{{x_{3}+x_{4}\neq
0}}$$x_{3}+x_{4}=0$$x_{3}=0$$x_{3}\neq 0$$x_{3}=0$$x_{3}\neq
0$$x_{2}=0$$x_{2}\neq
0$$x_{3}-x_{4}=0$$x_{3}+x_{4}=0$$x_{2}x_{5}+x_{4}^{2}=0$$x_{2}x_{5}+x_{4}^{2}\neq
0$$\stackrel{{\scriptstyle
k\notin\\{-1,0,1\\}}}{{x_{3}-kx_{4}=0}}$$x_{1}=0$$x_{1}\neq
0$$x_{1}=0$$x_{1}\neq 0$$x_{2}=0$$x_{2}\neq 0$$x_{1}=0$$x_{1}\neq
0$0IIextIIIIIIVVVI${\rm V}^{{\rm ext}}$${\rm VI}^{{\rm
ext}}$VII|k|∉{0,1}VIIIIX$x_{6}\neq
0$$x_{5}=0$$\stackrel{{\scriptstyle\tiny{\rm No\,solutions}}}{{x_{5}\neq
0}}$$\stackrel{{\scriptstyle\tiny{\rm No\,solutions}}}{{-x_{3}+x_{4}\neq
0}}$$-x_{3}+x_{4}=0$$x_{1}x_{6}+x_{4}^{2}=0$$x_{1}x_{6}+x_{4}^{2}\neq
0$VIIIext${\rm IX}^{{\rm ext}}$

The connected parts of the subsets denoted in the above Darboux tree are the
orbits of ${\rm Aut}_{c}(\mathfrak{s}_{6})$.

The Lie algebra automorphism group of $\mathfrak{s}_{6}$ reads

${\rm
Aut}(\mathfrak{s}_{6})=\left\\{\left(\begin{array}[]{cccc}-T^{3}_{2}T^{2}_{3}&T^{4}_{2}T_{3}^{2}&T^{4}_{3}T^{3}_{2}&T^{4}_{1}\\\
0&0&T^{3}_{2}&T^{4}_{2}\\\ 0&T^{2}_{3}&0&T^{4}_{3}\\\
0&0&0&-1\end{array}\right),\left(\begin{array}[]{cccc}T^{2}_{2}T^{3}_{3}&-T^{2}_{2}T^{4}_{3}&-T^{4}_{2}T^{3}_{3}&T^{4}_{1}\\\
0&T^{2}_{2}&0&T^{4}_{2}\\\ 0&0&T^{3}_{3}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):\begin{array}[]{c}T^{2}_{3},T^{2}_{2}\in\mathbb{R}\backslash\\{0\\},\\\
T^{3}_{2},T^{3}_{3}\in\mathbb{R}\backslash\\{0\\},\\\
T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\end{array}\right\\}.$

One element of each connected component of ${\rm Aut}(\mathfrak{s}_{6})$,
which are eight, and their lifts to $\Lambda^{2}\mathfrak{s}_{6}$ are given by

$\left(\begin{array}[]{cccc}\mp\lambda_{1}\lambda_{2}&0&0&0\\\ 0&\theta(\mp
1)\lambda_{1}&\theta(\pm 1)\lambda_{2}&0\\\ 0&\theta(\pm
1)\lambda_{1}&\theta(\mp 1)\lambda_{2}&0\\\ 0&0&0&\mp
1\end{array}\right)\Rightarrow{\small\left(\begin{array}[]{cccccc}\theta(\mp
1)\lambda_{2}&-\theta(\pm 1)\lambda_{1}&0&0&0&0\\\ -\theta(\pm
1)\lambda_{2}&\theta(\mp 1)\lambda_{1}&0&0&0&0\\\
0&0&\lambda_{1}\lambda_{2}&0&0&0\\\ 0&0&0&\mp\lambda_{1}\lambda_{2}&0&0\\\
0&0&0&0&\theta(\mp 1)\lambda_{1}&-\theta(\pm 1)\lambda_{2}\\\
0&0&0&0&-\theta(\pm 1)\lambda_{1}&\theta(\mp 1)\lambda_{2}\\\
\end{array}\right)},$ (7.6)

where $\theta(x)$ stands for the Heaviside function and
$\lambda_{1},\lambda_{2}\in\\{1,-1\\}$. By using the previous information, we
can verify whether some of the orbits of the action of ${\rm
Aut}_{c}(\mathfrak{s}_{6})$ on $\mathcal{Y}_{\mathfrak{s}_{6}}$ can be
connected by an element of ${\rm Aut}(\mathfrak{s}_{6})$, which gives rise to
the classes of equivalent $r$-matrices up to the action of Lie algebra
automorphisms of $\mathfrak{s}_{6}$. Since
$(\Lambda^{2}\mathfrak{s}_{6})^{\mathfrak{s}_{6}}=0$, each class of equivalent
$r$-matrices gives rise to a separate class of coboundary Lie bialgebras on
$\mathfrak{s}_{6}$. Hence, the classes of equivalent coboundary Lie bialgebras
are given by

${a)}\,{\rm I},{\rm I}^{\rm ext}\,{b)}\,{\rm II},\,{c)},\,{\rm
III},\,{d)}\,{\rm IV},\,{e)}\,{\rm V},{\rm V}^{\rm ext}\,{f})\,{\rm
VI,VI^{ext}},\,{g)}\,{\rm VII}_{|k|\neq\\{0,1\\}},\,{h)}\,{\rm
VIII,VIII^{ext}},\,{i)}\,{\rm IX,IX^{ext}}.$

###  7.7 Lie algebra $\mathfrak{s}_{7}$

The structure constants of Lie algebra $\mathfrak{s}_{7}$ are given in Table
1. Using this information, we can compute some relevant Schouten brackets
between the elements of the bases of $\mathfrak{s}_{7}$,
$\Lambda^{2}\mathfrak{s}_{7}$, and $\Lambda^{3}\mathfrak{s}_{7}$ (see Tables
3–5). Note that, from these tables, we get
$(\Lambda^{2}\mathfrak{s}_{7})^{\mathfrak{s}_{7}}=0$ and
$(\Lambda^{3}\mathfrak{s}_{7})^{\mathfrak{s}_{7}}=\langle e_{123}\rangle$.

By Remark 2.1, one obtains that

$\mathfrak{der}(\mathfrak{s}_{7}):=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&\mu_{13}&\mu_{14}\\\