0&\frac{1}{2}\mu_{11}&\mu_{23}&\mu_{12}\\\
0&-\mu_{23}&\frac{1}{2}\mu_{11}&\mu_{13}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{13},\mu_{14},\mu_{23}\in\mathbb{R}\right\\},$

which give rise to the basis of $V_{\mathfrak{s}_{7}}$ of the form

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

Then,

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

Since $(\Lambda^{3}\mathfrak{s}_{7})^{\mathfrak{s}_{7}}=\langle
e_{123}\rangle$, then the mCYBE reads

$x_{5}=x_{6}=0.$

Meanwhile, the CYBE takes the form

$x_{4}=x_{5}=x_{6}=0.$

The Darboux tree is presented below.

$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$$x_{3}=0$$x_{4}-kx_{3}=0$$x_{1}^{2}+x_{2}^{2}=0$$x_{1}^{2}+x_{2}^{2}\neq
0$0IIIIII${\rm IV}_{k\neq 0}$

The connected parts of the subsets denoted in the above Darboux tree are the
orbits of ${\rm Aut}_{c}(\mathfrak{s}_{7})$ in
$\mathcal{Y}_{\mathfrak{s}_{7}}$. By using the extension of the automorphism
group ${\rm Aut}(\mathfrak{s}_{7})$ to $\Lambda^{2}\mathfrak{s}_{7}$, we can
verify whether some of these parts are additionally connected by a Lie algebra
automorphism of $\mathfrak{s}_{7}$. The results are summarised in Table 2. As
before, each family of $r$-matrices induces a separate class of coboundary
coproducts.

The Lie algebra automorphism group of $\mathfrak{s}_{7}$ reads

$\begin{split}{\rm
Aut}(\mathfrak{s}_{7})&=\left\\{\left(\begin{array}[]{cccc}\pm[(T^{2}_{2})^{2}+(T^{2}_{3})^{2}]&\pm
T^{4}_{2}T^{2}_{2}-T^{4}_{3}T^{2}_{3}&T^{4}_{3}T^{2}_{2}\pm
T^{4}_{2}T^{2}_{3}&T^{4}_{1}\\\ 0&T^{2}_{2}&T^{2}_{3}&T^{4}_{2}\\\ 0&\mp
T^{2}_{3}&\pm T^{2}_{2}&T^{4}_{3}\\\ 0&0&0&\pm
1\end{array}\right):\begin{array}[]{c}(T^{2}_{2})^{2}+(T_{3}^{2})^{2}\in\mathbb{R}\backslash\\{0\\}\\\
T^{2}_{2},T^{2}_{3},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\end{array}\right\\}.\end{split}$

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

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

can be parametrised via $\phi\in[0,2\pi[$ and
$\mu:=[(T^{2}_{2})^{2}+(T^{2}_{3})^{2}]^{1/2}\in\mathbb{R}_{+}$ by setting
$T^{2}_{2}=\mu\cos\phi$ and $T^{2}_{3}=\mu\sin\phi$, one gets that ${\rm
Aut}(\mathfrak{s}_{7})$ has two connected components. As usual, we only need
one element for every connected component of ${\rm Aut}(\mathfrak{s}_{7})$ and
their lifts to $\Lambda^{2}\mathfrak{s}_{7}$, namely

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

Taking this into account and since
$(\Lambda^{2}\mathfrak{s}_{7})^{\mathfrak{s}_{7}}=0$, we obtain that the
families or equivalent coboundary Lie bialgebras are given by

${\rm a)}\,{\rm I},\,{\rm b)}\,{\rm II}_{+},\,{\rm c)}\,{\rm II}_{-},\,{\rm
d)}\,{\rm III},\,{\rm e)}\,({\rm IV}_{+})_{|k|\in\mathbb{R}_{+}}\,{\rm
f)}\,({\rm IV}_{-})_{|k|\in\mathbb{R}_{+}}.$

###  7.8 Lie algebra $\mathfrak{s}_{8}$

The structure constants for the Lie algebra $\mathfrak{s}_{8}$ are given in
Table 1. Recall that $\alpha\in]-1,1]\backslash\\{0\\}$. As in the previous
cases, one can obtain some relevant Schouten brackets between the basis
elements of $\mathfrak{s}_{8}$, $\Lambda^{2}\mathfrak{s}_{8}$, and
$\Lambda^{3}\mathfrak{s}_{8}$ (see Tables 3–5). In view of Tables 1,3–5, one
has that $(\Lambda^{2}\mathfrak{s}_{8})^{\mathfrak{s}_{8}}=\langle
e_{13}\rangle$ for $\alpha=-\frac{1}{2}$ and
$(\Lambda^{2}\mathfrak{s}_{8})^{\mathfrak{s}_{8}}=0$, otherwise. Moreover,
$(\Lambda^{3}\mathfrak{s}_{8})^{\mathfrak{s}_{8}}=0$.

By Remark 2.1, one obtains that the derivations of
$\mathfrak{s}^{\alpha}_{8}$, for a fixed value
$\alpha\in]-1,1[\backslash\\{0\\}$ read

$\mathfrak{der}(\mathfrak{s}^{\alpha}_{8})=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&\mu_{13}&\mu_{14}\\\
0&\mu_{22}&0&-\mu_{13}\\\ 0&0&\mu_{11}-\mu_{22}&\alpha\mu_{12}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{13},\mu_{14},\mu_{22}\in\mathbb{R}\right\\}.$
(7.7)

In case $\alpha=1$, one gets

$\mathfrak{der}(\mathfrak{s}^{1}_{8})=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&\mu_{13}&\mu_{14}\\\
0&\mu_{22}&\mu_{23}&-\mu_{13}\\\ 0&\mu_{32}&\mu_{11}-\mu_{22}&\mu_{12}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{13},\mu_{14},\mu_{22},\mu_{23},\mu_{32}\in\mathbb{R}\right\\}.$
(7.8)

For any value of $\alpha\in]-1,1]\backslash\\{0\\}$ and an element
$r\in\Lambda^{2}\mathfrak{s}_{8}$, one obtains

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

Since $(\Lambda^{3}\mathfrak{s}_{8})^{\mathfrak{s}_{8}}=0$, the mCYBE reads

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

Let us now consider two cases given by $\alpha\in]-1,1[\backslash\\{0\\}$ and
$\alpha=1$. In the first case, the space of derivations gives rise to the
basis of $V_{\mathfrak{s}^{\alpha}_{8}}$ of the form

$\begin{gathered}X_{1}=x_{1}\partial_{x_{1}}+2x_{2}\partial_{x_{2}}+x_{3}\partial_{x_{3}}+x_{4}\partial_{x_{4}}+x_{6}\partial_{x_{6}},\qquad
X_{2}=(x_{3}\alpha+x_{4})\partial_{x_{2}}+x_{5}\partial_{x_{3}}+\alpha
x_{5}\partial_{x_{4}},\\\
X_{3}=-(x_{3}+x_{4})\partial_{x_{1}}+x_{6}\partial_{x_{3}}+x_{6}\partial_{x_{4}},\qquad
X_{4}=-x_{5}\partial_{x_{1}}-x_{6}\partial_{x_{2}},\qquad
X_{5}=x_{1}\partial_{x_{1}}-x_{2}\partial_{x_{2}}+x_{5}\partial_{x_{5}}-x_{6}\partial_{x_{6}}.\end{gathered}$
(7.9)

The following diagram depicts the Darboux families that give us the orbits of
${\rm Aut}_{c}(\mathfrak{s}^{\alpha}_{8})$ on
$\mathcal{Y}_{\mathfrak{s}^{\alpha}_{8}}$. Since $x_{5},x_{6}$ are the bricks
for every $\mathfrak{s}^{\alpha}_{8}$, our Darboux tree starts with the 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$$x_{4}-\alpha
x_{3}=0$$\stackrel{{\scriptstyle\rm No\,\,solutions}}{{x_{4}-\alpha x_{3}\neq
0}}$$x_{3}=0$$x_{3}\neq 0$$\stackrel{{\scriptstyle(x_{3}\neq
0)}}{{x_{3}+x_{4}\neq 0}}$$\stackrel{{\scriptstyle{\rm
No\,\,solutions}}}{{x_{3}+x_{4}=0}}$$\alpha
x_{3}^{2}-x_{2}x_{5}=0$$\stackrel{{\scriptstyle(\alpha=-1/2)}}{{\alpha
x_{3}^{2}-x_{2}x_{5}\neq 0}}$$x_{2}=0$$x_{2}\neq
0$$(1+\alpha)x_{3}+x_{4}=0$$\stackrel{{\scriptstyle{\rm
No\,\,solutions}}}{{(1+\alpha)x_{3}+x_{4}\neq 0}}$$x_{1}=0$$x_{1}\neq
0$$x_{1}=0$$x_{1}\neq 0$0IIIIIIIVVVIVIIα=-1/2$x_{6}\neq
0$$x_{5}=0$$\stackrel{{\scriptstyle(\alpha=1)}}{{x_{5}\neq
0}}$$x_{3}-x_{4}=0$$\stackrel{{\scriptstyle{\rm
No\,\,solutions}}}{{x_{3}-x_{4}\neq
0}}$$x_{4}-x_{3}=0$$\stackrel{{\scriptstyle\rm
No\,\,solutions}}{{x_{4}-x_{3}\neq 0}}$$\stackrel{{\scriptstyle{\rm
No\,\,solutions}}}{{x_{3}^{2}+x_{1}x_{6}\neq
0}}$$x_{3}^{2}+x_{1}x_{6}=0$$x_{1}x_{6}-x_{2}x_{5}+x_{3}^{2}=0$$\stackrel{{\scriptstyle{\rm
No\,\,solutions}}}{{x_{1}x_{6}-x_{2}x_{5}+x_{3}^{2}\neq 0}}$VIIIIXα=1

The Lie algebra automorphism group of $\mathfrak{s}^{\alpha}_{8}$ for each
$\alpha\in]-1,1[\backslash\\{0\\}$, reads

${\rm
Aut}(\mathfrak{s}^{\alpha}_{8})=\left\\{\left(\begin{array}[]{cccc}T^{2}_{2}T^{3}_{3}&T^{2}_{1}&T^{3}_{1}&T^{4}_{1}\\\
0&T^{2}_{2}&0&-\frac{T^{3}_{1}}{T^{3}_{3}}\\\ 0&0&T^{3}_{3}&\frac{\alpha
T^{2}_{1}}{T^{2}_{2}}\\\
0&0&0&1\end{array}\right):\begin{array}[]{c}T^{2}_{2},\,T^{3}_{3}\in\mathbb{R}\backslash\\{0\\}\\\
T^{2}_{1},T^{3}_{1},T^{4}_{1}\in\mathbb{R}\end{array}\right\\}.$

In reality, we are only concerned with obtaining one element of ${\rm
Aut}(\mathfrak{s}^{\alpha}_{8})$ and its lift to $\Lambda^{2}\mathfrak{s}_{8}$
for each one of its connected components. For instance, we can choose

$T_{\lambda_{1},\lambda_{2}}=\left(\begin{array}[]{cccc}\lambda_{1}\lambda_{2}&0&0&0\\\
0&\lambda_{1}&0&0\\\ 0&0&\lambda_{2}&0\\\
0&0&0&1\end{array}\right)\quad\Longrightarrow\quad\Lambda^{2}T_{\lambda_{1},\lambda_{2}}=\left(\begin{array}[]{cccccc}\lambda_{2}&0&0&0&0&0\\\
0&\lambda_{1}&0&0&0&0\\\ 0&0&\lambda_{1}\lambda_{2}&0&0&0\\\
0&0&0&\lambda_{1}\lambda_{2}&0&0\\\ 0&0&0&0&\lambda_{1}&0\\\
0&0&0&0&0&\lambda_{2}\end{array}\right),\quad\lambda_{1},\lambda_{2}\in\\{\pm
1\\}.$ (7.10)

As in previous sections, the maps $\Lambda^{2}T_{\lambda_{1},\lambda_{2}}$
allow us to identify the orbits of the action of ${\rm
Aut}(\mathfrak{s}^{\alpha}_{8})$ on $\mathcal{Y}_{\mathfrak{s}^{\alpha}_{8}}$.
Our results are presented in Table 2. Moreover, for $\alpha\neq-\frac{1}{2}$,
each family of equivalent $r$-matrices give rise to a separate class of
equivalent coboundary Lie bialgebras. For $\alpha=-\frac{1}{2}$, we obtain
eight families of equivalent coboundary Lie bialgebras given by:

$a)\,{\rm II}\,({\rm zero\,class}),\quad b)\,{\rm I,III},\quad c)\,{\rm
IV},\quad d)\,{\rm V},\quad e)\,{\rm VI},{\rm
VII}^{\alpha=-\frac{1}{2}}_{-},{\rm VII}^{\alpha=-\frac{1}{2}}_{+},\quad
f)\,{\rm VIII}.$

Let us know study the case $\alpha=1$ using the previous results. Since (7.7)
for $\alpha=1$ is a Lie subalgebra of the space of derivations for
$\mathfrak{s}_{8}^{\alpha}$ given in (7.8), the Lie algebra spanned by the
vector fields of (7.9) for $\alpha=1$ is a Lie subalgebra of the Lie algebra
of fundamental vector fields of the action of ${\rm
Aut}(\mathfrak{s}_{8}^{1})$ on $\Lambda^{2}\mathfrak{s}_{8}$. Hence, the loci
of the above Darboux families allow us to characterise the strata of the
distribution spanned by the vector fields (7.9) for $\alpha=1$, which in turn
gives as the orbits of a Lie subgroup of ${\rm Aut}(\mathfrak{s}_{8}^{1})$. To
easily follow our discussion, we detail that

${\rm
Aut}(\mathfrak{s}^{\alpha}_{8})=\left\\{\left(\begin{array}[]{cccc}T^{2}_{2}T^{3}_{3}-T_{2}^{3}T^{2}_{3}&-T_{2}^{4}T_{3}^{2}+T^{2}_{2}T_{3}^{4}&-T_{2}^{4}T^{3}_{3}+T_{2}^{3}T_{3}^{4}&T^{4}_{1}\\\
0&T^{2}_{2}&T_{2}^{3}&T_{2}^{4}\\\ 0&T_{3}^{2}&T^{3}_{3}&T_{3}^{4}\\\
0&0&0&1\end{array}\right):\begin{array}[]{c}T^{2}_{2}T^{3}_{3}-T_{3}^{2}T^{3}_{2}\neq
0,\,T_{1}^{4},T_{2}^{4},T_{3}^{4}\in\mathbb{R}\end{array}\right\\}.$

To obtain the orbits of ${\rm Aut}(\mathfrak{s}_{8}^{1})$ on
$\mathcal{Y}_{\mathfrak{s}_{8}^{1}}$, it is enough to write ${\rm
Aut}(\mathfrak{s}_{8}^{1})$ as a composition of the previous subgroup with
certain Lie algebra automorphisms of ${\rm Aut}(\mathfrak{s}_{8}^{1})$, e.g.
the Lie algebra automorphisms $T_{A}=(\det A){\rm Id}\otimes A\otimes{\rm Id}$
for every $A\in GL(2,\mathbb{R})$. Hence, $\Lambda^{2}T_{A}=(\det
A)A\otimes(\det A){\rm Id}\otimes(\det A){\rm Id}$. The action of the
$\Lambda^{2}T_{A}$ on the loci of the Darboux families of (7.9) assuming
$\alpha=1$ allows us to obtain the orbits of the action of ${\rm
Aut}(\mathfrak{s}_{8}^{1})$ on $\mathcal{Y}_{\mathfrak{s}_{8}^{1}}$. In
particular, one obtains the subsets given in Table 2. Since
$(\Lambda^{2}\mathfrak{s}^{1}_{8})^{\mathfrak{s}^{1}_{8}}=0$, the classes of
equivalent coboundary coproducts for $\mathfrak{s}_{8}^{1}$ are given by

$a)\,\mathcal{I},\quad b)\,\mathcal{II},\quad c)\,\mathcal{III},\quad
d)\,\mathcal{IV}.$

###  7.9 Lie algebra $\mathfrak{s}_{9}$

As previously, we use the structure constants for $\mathfrak{s}_{9}$ in Table
1 to compute the Schouten brackets given in Tables 3–5. Moreover,
$(\Lambda^{2}\mathfrak{s}_{9})^{\mathfrak{s}_{9}}=0$ and
$(\Lambda^{3}\mathfrak{s}_{9})^{\mathfrak{s}_{9}}=0$ for every $\alpha>0$.
These calculations are enough to verify the remaining results of present
subsection.

By Remark 2.1, one obtains that

$\mathfrak{der}(\mathfrak{s}^{\alpha}_{9}):=\left\\{\left(\begin{array}[]{cccc}2\mu_{22}&\mu_{12}&\mu_{13}&\mu_{14}\\\
0&\mu_{22}&\mu_{23}&\mu_{12}-\alpha\mu_{13}\\\
0&-\mu_{23}&\mu_{22}&\alpha\mu_{12}+\mu_{13}\\\
0&0&0&0\end{array}\right):\mu_{12},\mu_{13},\mu_{14},\mu_{22},\mu_{23}\in\mathbb{R}\right\\}.$

The obtained derivations give rise to the basis of
$V_{\mathfrak{s}^{\alpha}_{9}}$ of the form

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

For an element $r\in\Lambda^{2}\mathfrak{s}_{9}$, we get

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

And since $(\Lambda^{3}\mathfrak{s}_{9})^{\mathfrak{s}_{9}}=0$ for every value
of $\alpha$, the mCYBE and the CYBE are equal and read

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

The Darboux tree for the class $\mathfrak{s}_{9}$ is presented below.

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

If we define $\Delta:=(T^{2}_{2})^{2}+(T^{2}_{3})^{2}$, the Lie algebra
automorphisms group of each $\mathfrak{s}^{\alpha}_{9}$ reads

${\rm
Aut}(\mathfrak{s}^{\alpha}_{9})=\left\\{\left(\begin{array}[]{cccc}\Delta&\frac{T^{4}_{2}(T^{2}_{2}+\alpha
T^{2}_{3})+T^{4}_{3}(\alpha T^{2}_{2}-T^{2}_{3})}{1+\alpha^{2}}&\frac{(-\alpha
T^{2}_{2}+T^{2}_{3})T^{4}_{2}+T^{4}_{3}(T^{2}_{2}+\alpha
T^{2}_{3})}{1+\alpha^{2}}&T^{4}_{1}\\\ 0&T^{2}_{2}&T^{2}_{3}&T^{4}_{2}\\\
0&-T^{2}_{3}&T^{2}_{2}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):\begin{array}[]{c}\Delta\in\mathbb{R}_{+},\\\
T^{2}_{2},T^{2}_{3},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\end{array}\right\\}.$

Using ideas from Section 7.5, we obtain that each ${\rm
Aut}(\mathfrak{s}^{\alpha}_{9})$ has one connected component. Consequently,
the orbits of ${\rm Aut}(\mathfrak{s}^{\alpha}_{9})$ on
$\Lambda^{2}\mathfrak{s}^{\alpha}_{9}$ are the strata of
$\mathscr{E}_{\mathfrak{s}^{\alpha}_{9}}$. Since
$(\Lambda^{2}\mathfrak{s}_{9})^{\mathfrak{s}_{9}}=0$ for every $\alpha>0$, the
strata of $\mathscr{E}_{\mathfrak{s}^{\alpha}_{9}}$ within
$\mathcal{Y}_{\mathfrak{s}^{\alpha}_{9}}$ amount for the families of
equivalent coboundary Lie bialgebras on $\mathfrak{s}^{\alpha}_{9}$. Our final
results are summarised in Table 2.

###  7.10 Lie algebra $\mathfrak{s}_{10}$

As in the previous cases, we hereafter use the structure constants for Lie
algebra $\mathfrak{s}_{10}$ in Table 1 and the induced Schouten brackets
between elements of $\mathfrak{s}_{10}$, $\Lambda^{2}\mathfrak{s}_{10}$, and
$\Lambda^{3}\mathfrak{s}_{10}$ depicted in Tables 3–5. It is remarkable that
$(\Lambda^{3}\mathfrak{s}_{10})^{\mathfrak{s}_{10}}=0$ and
$(\Lambda^{2}\mathfrak{s}_{10})^{\mathfrak{s}_{10}}=0$.

By Remark 2.1, the derivations of $\mathfrak{s}_{10}$ read

$\mathfrak{der}(\mathfrak{s}_{10})=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&\mu_{13}&\mu_{14}\\\
0&\frac{1}{2}\mu_{11}&\mu_{23}&\mu_{12}-\mu_{13}\\\
0&0&\frac{1}{2}\mu_{11}&\mu_{12}\\\
0&0&0&0\end{array}\right):\mu_{11},\mu_{12},\mu_{13},\mu_{14},\mu_{23}\in\mathbb{R}\right\\},$

which give rise to the basis of $V_{\mathfrak{s}_{10}}$ of the form

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

For an element $r\in\Lambda^{2}\mathfrak{s}_{10}$, we get

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

Since $(\Lambda^{3}\mathfrak{s}_{10})^{\mathfrak{s}_{10}}=0$, the mCYBE and
the CYBE are equal and they read

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

Since $x_{6}$ is the only brick for $\mathfrak{s}_{10}$, our Darboux tree
starts with the cases $x_{6}=0$ and $x_{6}\neq 0$. The full Darboux tree is
presented below.

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

The connected parts of the subspaces denoted in the Darboux tree are the
orbits of ${\rm Aut}_{c}(\mathfrak{s}_{10})$ within
$\mathcal{Y}_{\mathfrak{s}_{10}}$. To obtain the orbits of ${\rm
Aut}(\mathfrak{s}_{10})$ within $\mathcal{Y}_{\mathfrak{s}_{10}}$, we proceed
as in previous sections and we derive the automorphism group of
$\mathfrak{s}_{10}$, which takes the form

${\rm
Aut}(\mathfrak{s}_{10})=\left\\{\left(\begin{array}[]{cccc}(T^{2}_{2})^{2}&T^{4}_{3}T^{2}_{2}&T^{4}_{3}T^{2}_{2}+T^{4}_{3}T^{3}_{2}-T^{4}_{2}T^{2}_{2}&T^{4}_{1}\\\
0&T^{2}_{2}&T^{3}_{2}&T^{4}_{2}\\\ 0&0&T^{2}_{2}&T^{4}_{3}\\\
0&0&0&1\end{array}\right):\begin{array}[]{c}T^{2}_{2}\in\mathbb{R}\backslash\\{0\\},\\\
T^{3}_{2},T^{4}_{1},T^{4}_{2},T^{4}_{3}\in\mathbb{R}\end{array}\right\\}.$

In reality, we do not need the full group. It is enough to note that there are
two connected components of ${\rm Aut}(\mathfrak{s}_{10})$, represented by two
automorphisms that read, along with their extensions to
$\Lambda^{2}\mathfrak{s}_{10}$, as follows

$T_{\lambda}:=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&\lambda&0&0\\\
0&0&\lambda&0\\\
0&0&0&1\end{array}\right)\qquad\Longrightarrow\qquad\Lambda^{2}T_{\lambda}:={\small\left(\begin{array}[]{cccccc}\lambda&0&0&0&0&0\\\
0&\lambda&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&\lambda&0\\\
0&0&0&0&0&\lambda\\\ \end{array}\right)},\quad\lambda\in\\{\pm 1\\}.$ (7.11)

By applying $\Lambda^{2}T_{\lambda}$ to the orbits of ${\rm
Aut}_{c}(\mathfrak{s}_{10})$ in $\mathcal{Y}_{\mathfrak{s}_{10}}$, depicted in
Table 2, we obtain the orbits of Aut($\mathfrak{s}_{10})$ in
$\mathcal{Y}_{\mathfrak{s}_{10}}$. Since
$(\Lambda^{2}\mathfrak{s}_{10})^{\mathfrak{s}_{10}}=0$, such orbits amount to
the classification of equivalent coboundary Lie bialgebras up to Lie algebra
automorphisms of $\mathfrak{s}_{10}$.

###  7.11 Lie algebra $\mathfrak{s}_{11}$

As previously, the information in Tables 1, 3–5 allows us to accomplish the
calculations of this section. In particular, one gets
$(\Lambda^{2}\mathfrak{s}_{11})^{\mathfrak{s}_{11}}=0$ and
$(\Lambda^{3}\mathfrak{s}_{11})^{\mathfrak{s}_{11}}=0$.

By Remark 2.1, one obtains

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

Such derivations give rise to the basis of $V_{\mathfrak{s}_{11}}$ of the form

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

For an element $r\in\Lambda^{2}\mathfrak{s}_{11}$, we get

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

Since $(\Lambda^{3}\mathfrak{s}_{11})^{\mathfrak{s}_{11}}=\\{0\\}$, the mCYBE
and the CYBE are equal and read

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

Since $x_{5},x_{6}$ are the bricks for $\mathfrak{s}_{11}$, our Darboux tree
starts with the cases $x_{i}=0$ and $x_{i}\neq 0$, for $i\in\\{5,6\\}$. The
Darboux tree is presented below.

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

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

The above Darboux tree gives the orbits of ${\rm Aut}_{c}(\mathfrak{s}_{11})$
in $\mathcal{Y}_{\mathfrak{s}_{11}}$. To obtain the equivalence classes of
$r$-matrices up to Lie algebra automorphisms of ${\rm
Aut}(\mathfrak{s}_{11})$, we obtain that

${\rm
Aut}(\mathfrak{s}_{11})=\left\\{\left(\begin{array}[]{cccc}T^{2}_{2}T^{3}_{3}&T^{2}_{1}&-T^{3}_{3}T^{4}_{2}&T^{4}_{1}\\\
0&T^{2}_{2}&0&T^{4}_{2}\\\ 0&0&T^{3}_{3}&0\\\
0&0&0&1\end{array}\right):\begin{array}[]{c}T^{2}_{2},T^{3}_{3}\in\mathbb{R}\backslash\\{0\\}\\\
T^{2}_{1},T^{4}_{1},T^{4}_{2}\in\mathbb{R}\end{array}\right\\}.$

In fact, only one representative for each connected component is needed for
our purposes. There are four connected components of ${\rm
Aut}(\mathfrak{s}_{11})$, each one represented by one of the following
automorphisms, which are given together with their lifts to
$\Lambda^{2}\mathfrak{s}_{11}$, namely

$T_{\lambda_{1},\lambda_{2}}:=\left(\begin{array}[]{cccc}\lambda_{1}\lambda_{2}&0&0&0\\\
0&\lambda_{1}&0&0\\\ 0&0&\lambda_{2}&0\\\
0&0&0&1\end{array}\right)\quad\Longrightarrow\quad\Lambda^{2}T_{\lambda_{1},\lambda_{2}}={\small\left(\begin{array}[]{cccccc}\lambda_{2}&0&0&0&0&0\\\
0&\lambda_{1}&0&0&0&0\\\ 0&0&\lambda_{1}\lambda_{2}&0&0&0\\\
0&0&0&\lambda_{1}\lambda_{2}&0&0\\\ 0&0&0&0&\lambda_{1}&0\\\
0&0&0&0&0&\lambda_{2}\\\
\end{array}\right),\qquad\lambda_{1},\lambda_{2}\in\\{\pm 1\\}}.$ (7.12)

The $\Lambda^{2}T_{\lambda_{1},\lambda_{2}}$ allow us to check whether the
orbits of ${\rm Aut}_{c}(\mathfrak{s}_{11})$ are connected by an element of
${\rm Aut}(\mathfrak{s}_{11})$ acting on $\Lambda^{2}\mathfrak{s}_{11}$. The
equivalence classes of such $r$-matrices (up to the action of elements of
${\rm Aut}(\mathfrak{s}_{11})$, as standard) are given in Table 2. Since
$(\Lambda^{2}\mathfrak{s}_{11})^{\mathfrak{s}_{11}}=0$, each family of
$r$-matrices gives rise to a separate class of equivalent coboundary Lie
bialgebras on $\mathfrak{s}_{11}$.

###  7.12 Lie algebra $\mathfrak{s}_{12}$

The structure constants for Lie algebra $\mathfrak{s}_{12}$ are given in Table
1. This along with the selected Schouten brackets between basis elements of
$\mathfrak{s}_{12}$, $\Lambda^{2}\mathfrak{s}_{12}$, and
$\Lambda^{3}\mathfrak{s}_{12}$ described in Tables 3–5, allow us to accomplish
the calculations of this section. In particular, we obtain that
$(\Lambda^{2}\mathfrak{s}_{12})^{\mathfrak{s}_{12}}=0$ and
$(\Lambda^{3}\mathfrak{s}_{12})^{\mathfrak{s}_{12}}=0$.

By Remark 2.1, one obtains that

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

The previous derivations give rise to the basis of $V_{\mathfrak{s}_{12}}$
given by

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

For an element $r\in\Lambda^{2}\mathfrak{s}_{12}$, we get

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

Since $(\Lambda^{3}\mathfrak{s}_{12})^{\mathfrak{s}_{12}}=0$, the mCYBE and
CYBE are equal and read

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

Since $x_{6}$ is the only brick for $\mathfrak{s}_{12}$, our Darboux tree
starts with the cases $x_{6}=0$ and $x_{6}\neq 0$. The full Darboux tree is
given next.

$x_{6}=0$$\begin{array}[]{c}x_{2}-x_{4}=0\\\
x_{2}+x_{4}=0\end{array}$$x_{2}^{2}+x_{4}^{2}\neq
0$$\begin{array}[]{c}x_{3}-x_{5}=0\\\
x_{3}+x_{5}=0\end{array}$$\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{x_{3}^{2}+x_{5}^{2}\neq
0}}$$x_{3}^{2}+x_{5}^{2}=0$$x_{3}^{2}+x_{5}^{2}\neq 0$$x_{1}=0$$x_{1}\neq
0$$x_{2}x_{3}+x_{4}x_{5}=0$$\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{x_{2}x_{3}+x_{4}x_{5}\neq 0}}$$\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{-x_{2}x_{5}+x_{3}^{2}+x_{3}x_{4}+x_{5}^{2}\neq
0}}$$-x_{2}x_{5}+x_{3}^{2}+x_{3}x_{4}+x_{5}^{2}=0$0IIIIII$x_{6}-k=0$, ($k\neq
0$) $x_{2}+x_{5}=0\\\ x_{3}-x_{4}=0$ $\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{x_{2}+x_{5}\neq 0\lor x_{3}-x_{4}\neq
0}}$$x_{5}^{2}+x_{3}^{2}-x_{2}x_{5}+x_{3}x_{4}+2x_{1}x_{6}=0$$\stackrel{{\scriptstyle{\rm
No\,solutions}}}{{x_{5}^{2}+x_{3}^{2}-x_{2}x_{5}+x_{3}x_{4}+2x_{1}x_{6}\neq
0}}$IV|k|>0

The connected parts of the loci of the Darboux families of the Darboux tree
are the orbits of ${\rm Aut}_{c}(\mathfrak{s}_{12})$ within
$\mathcal{Y}_{\mathfrak{s}_{12}}$. These are given by the connected parts of
the subsets given in Table 2. Let us obtain the orbits of ${\rm
Aut}(\mathfrak{s}_{12})$ on $\mathcal{Y}_{\mathfrak{sl}_{12}}$.

The Lie algebra automorphism group of $\mathfrak{s}_{12}$ reads

${\rm
Aut}(\mathfrak{s}_{12})=\left\\{\left(\begin{array}[]{cccc}T^{1}_{1}&T^{2}_{1}&T^{3}_{1}&\pm
T^{3}_{2}\\\ \mp T^{2}_{1}&\pm T^{1}_{1}&T^{3}_{2}&\mp T^{3}_{1}\\\ 0&0&1&0\\\
0&0&0&\pm
1\end{array}\right):\begin{array}[]{c}(T_{1}^{1})^{2}+(T^{2}_{1})^{2}\in\mathbb{R}\backslash\\{0\\}\\\
T^{1}_{1},T^{2}_{1},T^{3}_{1},T^{3}_{2}\in\mathbb{R}\end{array}\right\\}.$

There are two connected components of ${\rm Aut}(\mathfrak{s}_{12})$. One such
element for each connected component of ${\rm Aut}(\mathfrak{s}_{12})$ and
their extensions to $\Lambda^{2}\mathfrak{s}_{12}$ are given by

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

By using $\Lambda^{2}T_{\pm}$ on the loci of the above Darboux tree, we obtain
the orbits of ${\rm Aut}(\mathfrak{s}_{12})$ that are given by the classes of
equivalent $r$-matrices detailed in Table 2. Since
$(\Lambda^{2}\mathfrak{s}_{12})^{\mathfrak{s}_{12}}=0$, each family of
equivalent $r$-matrices gives rise to a separate class of equivalent
coboundary Lie bialgebras.

###  7.13 Lie algebra $\mathfrak{n}_{1}$

The structure constants for the Lie algebra $\mathfrak{n}_{1}$ are given in
Table 1, while relevant Schouten brackets between the elements of bases of
$\mathfrak{n}_{1}$, $\Lambda^{2}\mathfrak{n}_{1}$, and
$\Lambda^{3}\mathfrak{n}_{1}$ to be used hereafter are displayed in Tables
3–5. From these calculations, one obtains that
$(\Lambda^{2}\mathfrak{n}_{1})^{\mathfrak{n}_{1}}=\langle e_{12}\rangle$ and
$(\Lambda^{3}\mathfrak{n}_{1})^{\mathfrak{n}_{1}}=\langle
e_{123},e_{124}\rangle$. By Remark 2.1, one has that the derivations222Note
that the dimension of the space of derivation matches the result given in the
WŠ classification of $\mathfrak{n}_{1}$ take the form

$\mathfrak{der}(\mathfrak{n}_{1})=\left\\{\left(\begin{array}[]{cccc}\mu_{11}&\mu_{12}&\mu_{13}&\mu_{14}\\\
0&\mu_{22}&\mu_{12}&\mu_{24}\\\ 0&0&2\mu_{22}-\mu_{11}&\mu_{34}\\\
0&0&0&\mu_{11}-\mu_{22}\end{array}\right):\mu_{11},\mu_{12},\mu_{13},\mu_{14},\mu_{22},\mu_{24},\mu_{34}\in\mathbb{R}\right\\},$

which, by lifting them to $\Lambda^{2}\mathfrak{n}_{1}$, give rise to the
basis of $V_{\mathfrak{n}_{1}}$ of the form

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

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

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

Thus, CYBE reads

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

Since $(\Lambda^{3}\mathfrak{n}_{1})^{\mathfrak{n}_{1}}=\langle
e_{123},e_{124}\rangle$, the mCYBE differs from the CYBE and reads

$x_{6}=0.$

The Darboux tree for this Lie algebra reads

$x_{6}=0$$x_{5}=0$$x_{5}\neq 0$$x_{4}=0$$x_{4}\neq
0$$x_{2}x_{5}-x_{3}x_{4}=0$$x_{2}x_{5}-x_{3}x_{4}\neq 0$$x_{3}=0$$x_{3}\neq
0$$x_{3}=0$$x_{3}\neq 0$$x_{2}=0$$x_{2}\neq 0$$x_{1}=0$$x_{1}\neq
0$0IIIIIIIVVVIVII

As in all previous sections, the orbits of Aut${}_{c}(\mathfrak{n}_{1}$) in
$\mathcal{Y}_{\mathfrak{n}_{1}}$ are given by the connected components of the
loci of the Darboux families of the above Darboux tree. Results can be found
in Table 2.

Let us again obtain the orbits of Aut$(\mathfrak{n}_{1})$ in
$\mathcal{Y}_{\mathfrak{n}_{1}}$. The automorphism group of $\mathfrak{n}_{1}$
takes the form

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

It has four connected components represented by the following Lie algebra
automorphisms, which are also lifted to $\Lambda^{2}\mathfrak{n}_{1}$:

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

The action of the $\Lambda^{2}T_{\lambda_{1},\lambda_{2}}$ on the loci of the
above Darboux tree gives the families of equivalent $r$-matrices given in
Table 2. Since $(\Lambda^{2}\mathfrak{n}_{1})^{\mathfrak{n}_{1}}=\langle
e_{12}\rangle$ and using the information in Table 2, we see that the families
of equivalent coboundary Lie bialgebras on $\mathfrak{n}_{1}$ are given by
orbits of ${\rm Aut}(\mathfrak{n}_{1})$ with the exception that the class I
and the zero class (given by the zero $r$-matrix) give the Lie bialgebra in
$\mathfrak{n}_{1}$ with a zero coproduct.

##  8 Conclusions and outlook

This work has devised a generalisation of the theory of Darboux polynomials to
determine and to classify up to Lie algebra automorphisms the coboundary real
Lie bialgebras over indecomposable real four-dimensional Lie algebras in a
geometric manner. As a byproduct, a technique for the matrix representation of
Lie algebras with non-trivial kernel has been developed. Such matrix
representations are frequently useful in calculations.

The procedures devised in this work are good enough to make affordable the
classification, up to Lie algebra automorphisms, of coboundary Lie bialgebras
on real and complex, of at least dimension five, indecomposable Lie algebras
via the Šnobl and Winternitz classification [50]. The case of two- and three-
dimensional Lie bialgebras can also be obtained. Moreover, a brief analysis of
the classification of coboundary coproducts on real four-dimensional
decomposable Lie algebras through our methods shows that their classification
relies partially on the classification of coboundary coproducts on three- and
two-dimensional Lie algebras, while the complexity of the procedure is
significantly easier than in the present work [25, 35]. We expect to tackle
this task in the future and to compare our results with previous works on the
topic [11, 35, 30]. Our techniques are also expected to be applied to other
higher-dimensional Lie algebras of particular types, e.g. five-dimensional
nilpotent or semi-simple Lie algebras. In all previously mentioned cases, we
aim to inspect new properties of the Darboux families used to study these
problems.

The determination of coboundary Lie bialgebras is relevant to the study of
Poisson Lie groups. In particular, it represents an initial step in the study
of Poisson Lie groups of dimension four, which would give rise to an extension
of the paper [3].

Our work has focused on the classification of coboundary Lie bialgebras on
indecomposable Lie algebras. Probably, one laborious task in the application
of our method is the determination of a representative of each connected part
of the Lie group of Lie algebra automorphisms of a Lie algebra. In the case of
semi-simple Lie algebras, this is much easier, as algebraic techniques based
on Dynkin diagrams and other results can be applied [13, 40]. It is left for
further works to study this problem and to search for coboundary Lie
bialgebras in the Lie algebra $\mathfrak{so}(3,2)$, which has applications in
the analysis of quantum gravity in 3+1 dimensions (see [9] and references
therein).

##  9 Acknowledgements

D. Wysocki acknowledges support from a grant financed by the University of
Warsaw (UW) and the Kartezjusz program of the Jagiellonian University and UW.
J. de Lucas acknowledges partial financial support from the NCN grant HARMONIA
2016/22/M/ST1/00542.

$\mathfrak{g}$ | Orbit | Dim | $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $x_{5}$ | $x_{6}$ | Repr. element
---|---|---|---|---|---|---|---|---|---
$\mathfrak{s}_{1}$ | ${\rm I}_{\pm}$ | 1 | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | 0 | 0 | $\pm e_{12}$
II | 1 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{13}$
${\rm III}_{\pm}$ | 2 | $\mathbb{R}_{\pm}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $\pm e_{12}+e_{13}$
IV | 2 | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{23}$
${\rm V}_{\pm}$ | 3 | $\mathbb{R}_{\pm}$ | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $\pm e_{12}+e_{23}$
VI | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
VII | 3 | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{34}$
${\rm VIII}_{\pm}$ | 4 | $\mathbb{R}_{\pm}$ | $\mathbb{R}$ | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $\pm e_{12}+e_{34}$
$\mathfrak{s}_{2}$ | ${\rm I}_{\pm}$ | 1 | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | 0 | 0 | $\pm e_{12}$
${\rm II}_{\pm}$ | 2 | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | 0 | $\pm e_{13}$
${\rm III}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
${\rm IV}_{\pm}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | 0 | $\mathbb{R}_{\pm}$ | 0 | 0 | $\pm e_{23}$
| I | 1 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | 0 | $e_{12}$
II | 1 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{13}$
III | 2 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{12}+e_{13}$
IV | 1 | 0 | 0 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{23}$
V | 2 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{12}+e_{23}$
$\mathfrak{s}^{\alpha,\beta}_{3}$ | VI | 2 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{13}+e_{23}$
${\rm differ}$ | ${\rm VII}_{\pm}$ | 3 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $\pm x_{1}x_{2}x_{4}>0$ | 0 | 0 | $\pm(e_{12}+e_{13}+e_{23})$
$\alpha,\beta,1$ | VIII | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
| ${\rm IX}_{\alpha+\beta\in\\{0,-1*\\}}$ | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{14}+e_{23}$
| X | 3 | $\mathbb{R}$ | 0 | 0 | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}$
| ${\rm XI}_{\alpha+\beta=-1}$ | 4 | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{13}+e_{24}$
| XIV | 3 | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{34}$
| ${\rm XV}^{\alpha=-1}_{\alpha+\beta=-1,*}$ | 4 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}$ | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{12}+e_{34}$
| $\mathcal{I}$ | 2 | $\mathbb{R}$ | $x_{1}^{2}+x_{2}^{2}\neq 0$ | $0$ | $0$ | $0$ | 0 | $e_{12}$
| $\mathcal{II}$ | 1 | $0$ | $0$ | $0$ | $\mathbb{R}\backslash\\{0\\}$ | $0$ | 0 | $e_{23}$
$\mathfrak{s}^{\alpha,\alpha}_{3}$ | $\mathcal{III}$ | 3 | $\mathbb{R}$ | $x_{1}^{2}+x_{2}^{2}\neq 0$ | $0$ | $\mathbb{R}\backslash\\{0\\}$ | $0$ | 0 | $e_{12}+e_{23}$
| $\mathcal{IV}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $0$ | $0$ | 0 | $e_{14}$
α≠1 | $\mathcal{V}^{*}_{\alpha=-1/2}$ | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | $0$ | 0 | $e_{14}+e_{23}$
| $\mathcal{VI}^{*}_{\alpha=-1/2}$ | 5 | $x_{2}x_{5}-x_{1}x_{6}\neq 0$ | $0$ | $\mathbb{R}$ | $\mathbb{R}$ | $x_{5}^{2}+x_{6}^{2}\neq 0$ | $e_{12}+e_{34}$
| $\mathcal{VII}$ | 4 | $x_{2}x_{5}-x_{1}x_{6}=0$ | $0$ | $\mathbb{R}$ | $\mathbb{R}$ | $x_{5}^{2}+x_{6}^{2}\neq 0$ | $e_{34}$
| $\mathscr{I}$ | 1 | $\mathbb{R}\backslash\\{0\\}$ | $0$ | $0$ | $0$ | $0$ | $0$ | $e_{12}$
| $\mathscr{II}$ | 2 | $0$ | $\mathbb{R}$ | $0$ | $x_{2}^{2}+x_{4}^{2}\neq 0$ | $0$ | $0$ | $e_{13}$
$\mathfrak{s}^{1,\beta}_{3}$ | $\mathscr{III}$ | 3 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}$ | $0$ | $x_{2}^{2}+x_{4}^{2}\neq 0$ | $0$ | $0$ | $e_{12}+e_{13}$
| $\mathscr{IV}$ | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $x_{3}^{2}+x_{5}^{2}\neq 0$ | $\mathbb{R}$ | $x_{2}x_{5}-x_{3}x_{4}=0$ | $0$ | $e_{13}$
β≠1 | $\mathscr{V}_{\beta=-1}$ | 5 | $\mathbb{R}$ | $\mathbb{R}$ | $x_{3}^{2}+x_{5}^{2}\neq 0$ | $\mathbb{R}$ | $x_{2}x_{5}-x_{3}x_{4}\neq 0$ | $0$ | $e_{14}+e_{23}$
| $\mathscr{VI}$ | 4 | $0$ | $\mathbb{R}$ | $0$ | $\mathbb{R}$ | $0$ | $\mathbb{R}\backslash\\{0\\}$ | $e_{34}$
| ${\rm i}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $0$ | $x_{1}^{2}+x_{2}^{2}+x_{4}^{2}\neq 0$ | $0$ | $0$ | $e_{12}$
$\mathfrak{s}^{1,1}_{3}$ | ${\rm ii}$ | 5 | $x_{3}x_{4}+\\!x_{1}x_{6}=\\!x_{2}x_{5}$ | $\mathbb{R}$ | $x_{3}x_{4}+x_{1}x_{6}=x_{2}x_{5}$ | $\mathbb{R}$ | $x_{3}^{2}+x_{5}^{2}+x_{6}^{2}\neq 0$ | $e_{13}+e_{34}$
$\mathfrak{s}^{\alpha}_{4}$ | I± | 1 | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | 0 | 0 | $\pm e_{12}$
II | 1 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{13}$
$\alpha\notin\\{0,1\\}$ | III± | 2 | $\mathbb{R}_{\pm}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $\pm e_{12}+e_{13}$
IV | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
V | 2 | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{23}$
VI± | 3 | $\mathbb{R}_{\pm}$ | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $\pm e_{12}+e_{23}$
${\rm VII}_{\alpha\in\\{-2^{*},-1\\}}$ | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{14}+e_{23}$
VIII | 3 | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{34}$
${\rm IX}^{\alpha=-2,*}_{\pm}$ | 4 | $\mathbb{R}_{\pm}$ | $\mathbb{R}$ | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $\pm e_{12}+e_{34}$
${\rm X}_{\alpha=1}$ | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $-\frac{x_{1}x_{6}}{x_{3}}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{14}+e_{34}$
$\mathfrak{s}^{1}_{4}$ | $\mathcal{I}_{\pm}$ | 2 | $\mathbb{R}_{\pm}$ | $\mathbb{R}$ | 0 | 0 | 0 | 0 | $\pm e_{12}$
$\mathcal{II}$ | 1 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{13}$
$\mathcal{III}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
$\mathcal{IV}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{23}$
$\mathcal{V}$ | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $x_{4}x_{3}-x_{1}x_{6}=0$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{14}+e_{34}$
$\mathfrak{s}_{5}$ | I | 1 | $\mathbb{R}$ | $x_{1}^{2}+x_{2}^{2}\neq 0$ | 0 | 0 | 0 | 0 | $e_{12}$
${\rm II}_{\pm}$ | 1 | 0 | 0 | 0 | $\mathbb{R}_{\pm}$ | 0 | 0 | $\pm e_{23}$
${\rm III}_{\pm}$ | 2 | $\mathbb{R}$ | $x_{1}^{2}+x_{2}^{2}\neq 0$ | 0 | $\mathbb{R}_{\pm}$ | 0 | 0 | $e_{12}\pm e_{23}$
IV | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
$({\rm V}_{\pm})^{\beta=0}_{{\alpha=-2\beta}*}$ | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}_{\pm}$ | 0 | 0 | $e_{14}\pm e_{23}$
$\mathfrak{s}_{6}$ | I | 1 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | 0 | $e_{12}$
Iext | 1 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 |
II | 2 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{12}+e_{13}$
III | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
IV∗ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{23}$
V∗ | 2 | $\mathbb{R}$ | 0 | $x_{4}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{14}+e_{23}$
${\rm V}^{{\rm ext}*}$ | 2 | 0 | $\mathbb{R}$ | $-x_{4}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 |

$\mathfrak{g}$ | Orbit | Dim. | $x_{1}$ | $x_{2}$ | $x_{3}$ | $x_{4}$ | $x_{5}$ | $x_{6}$ | Repr. element
---|---|---|---|---|---|---|---|---|---
$\mathfrak{s}_{6}$ |  VI∗ | 3 | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $x_{4}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{13}+e_{14}+e_{23}$
${\rm VI}^{{\rm ext}*}$ | 3 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}$ | $-x_{4}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 |
${\rm VII}^{*}_{|k|\notin\\{0,1\\}}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $kx_{4}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $ke_{14}+e_{23}$
VIII | 4 | $\mathbb{R}$ | $-x_{4}^{2}/x_{5}$ | $\mathbb{R}$ | $-x_{3}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}$
VIIIext | 3 | $\frac{-x_{4}^{2}}{x_{6}}$ | $\mathbb{R}$ | $\mathbb{R}$ | $x_{3}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ |
IX∗ | 4 | $\mathbb{R}$ | $x_{2}+x_{4}^{2}/x_{5}\neq 0$ | $\mathbb{R}$ | $-x_{3}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}+e_{13}$
${\rm IX}^{{\rm ext}*}$ | 4 | $x_{1}+\frac{x_{4}^{2}}{x_{6}}\neq 0$ | $\mathbb{R}$ | $\mathbb{R}$ | $x_{3}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ |
$\mathfrak{s}_{7}$ | I | 2 | $\mathbb{R}$ | $x_{1}^{2}+x_{2}^{2}\neq 0$ | 0 | 0 | 0 | 0 | $e_{12}$
${\rm II}_{\pm}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | $\pm e_{14}$
III∗ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{23}$
$({\rm IV}_{\pm})^{*}_{|k|\in\mathbb{R}_{+}}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | $kx_{3}$ | 0 | 0 | $\pm e_{14}\pm ke_{23}$
$\mathfrak{s}_{8}$ | I | 1 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | 0 | $e_{12}$
II | 1 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{13}$
III | 2 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{12}+e_{13}$
IV | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
V | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $-(1+\alpha)x_{3}$ | 0 | 0 | $e_{14}-(1+\alpha)e_{23}$
VI | 3 | $\mathbb{R}$ | $\frac{\alpha x_{3}^{2}}{x_{5}}$ | $\mathbb{R}$ | $\alpha x_{3}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}$
${\rm VII}^{\alpha=-1/2}_{\pm}$ | 4 | $\mathbb{R}$ | $x_{2}x_{5}-\alpha x_{3}^{2}\in\mathbb{R}_{\pm}$ | $\mathbb{R}$ | $\alpha x_{3}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{13}\pm e_{24}$
VIII | 3 | $-\frac{x_{3}^{2}}{x_{6}}$ | $\mathbb{R}$ | $\mathbb{R}$ | $x_{3}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{34}$
$\mathfrak{s}^{1}_{8}$ | $\mathcal{I}$ | 2 | $\mathbb{R}$ | $x_{1}^{2}+x_{2}^{2}\neq 0$ | 0 | 0 | 0 | 0 | $e_{12}$
$\mathcal{II}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
$\mathcal{III}$ | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $-2x_{3}$ | 0 | 0 | $e_{14}-2e_{23}$
$\mathcal{IV}$ | 4 | $x_{1}x_{6}-x_{2}x_{5}+x_{3}^{2}=0$ | $x_{3}$ | $\mathbb{R}$ | $x_{5}^{2}+x_{6}^{2}\neq 0$ | $e_{24}$
$\mathfrak{s}_{9}$ | I | 2 | $\mathbb{R}$ | $x_{1}^{2}+x_{2}^{2}\neq 0$ | 0 | 0 | 0 | 0 | $e_{12}$
II± | 2 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | $\pm e_{14}$
III± | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | $-2\alpha x_{3}$ | 0 | 0 | $\pm 2\alpha e_{23}\mp e_{14}$
$\mathfrak{s}_{10}$ | I | 1 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | 0 | $e_{12}$
II | 2 | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{13}$
III± | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | $\pm e_{14}$
IV± | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | $-2x_{3}$ | 0 | 0 | $\pm e_{14}\mp 2e_{23}$
V | 3 | $\mathbb{R}$ | $\frac{x_{3}^{2}}{x_{5}}$ | $\mathbb{R}$ | $x_{3}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}$
$\mathfrak{s}_{11}$ | I | 1 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | 0 | $e_{12}$
II | 1 | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{13}$
III | 2 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | $e_{12}+e_{13}$
IV | 2 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
V | 3 | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{13}+e_{14}$
VI | 2 | 0 | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $-x_{3}$ | 0 | 0 | $e_{14}-e_{23}$
VII | 3 | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $-x_{3}$ | 0 | 0 | $e_{12}+e_{14}-e_{23}$
VIII | 3 | $\mathbb{R}$ | 0 | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}$
IX | 3 | $-\frac{x_{3}^{2}}{x_{6}}$ | $\mathbb{R}$ | $\mathbb{R}$ | $x_{3}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | $e_{34}$
$\mathfrak{s}_{12}$ | I | 1 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | 0 | $e_{12}$
II | 3 | $\mathbb{R}$ | $\mathbb{R}$ | 0 | $x_{2}^{2}+x_{4}^{2}\neq 0$ | 0 | 0 | $e_{13}$
III | 3 | $\mathbb{R}$ | $x_{5}$ | $\mathbb{R}$ | $-x_{3}$ | $x_{3}^{2}+x_{5}^{2}\neq 0$ | 0 | $e_{13}+e_{24}$
IV|k|>0 | 2 | $-\frac{x_{2}^{2}+x_{3}^{2}}{x_{6}}$ | $\mathbb{R}$ | $\mathbb{R}$ | $x_{3}$ | $-x_{2}$ | $k$ | $ke_{34}$
$\mathfrak{n}_{1}$ | I | 1 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | 0 | 0 | $e_{12}$
II± | 2 | $\mathbb{R}$ | $\mathbb{R}_{\pm}$ | 0 | 0 | 0 | 0 | $\pm e_{13}$
III | 3 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | 0 | $e_{14}$
IV | 3 | $\mathbb{R}$ | $\mathbb{R}$ | 0 | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{23}$
V | 4 | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | 0 | $e_{14}+e_{23}$
VI∗ | 4 | $\mathbb{R}$ | $\frac{x_{3}x_{4}}{x_{5}}$ | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}$
VII${}_{\pm}^{*}$ | 5 | $\mathbb{R}$ | $x_{2}-\frac{x_{3}x_{4}}{x_{5}}\in\mathbb{R}_{\pm}$ | $\mathbb{R}$ | $\mathbb{R}$ | $\mathbb{R}\backslash\\{0\\}$ | 0 | $e_{24}\pm e_{13}$

Table 2: Orbits of the action of ${\rm Aut}(\mathfrak{g})$ on
$\mathcal{Y}_{\mathfrak{g}}$, their dimensions, elements, and representatives,
for real four-dimensional indecomposable Lie algebras $\mathfrak{g}$. Each Lie
algebra is divided into several subsets enumerated by roman numbers, which
classify the orbits of ${\rm Aut}(\mathfrak{g})$ in
$\mathcal{Y}_{\mathfrak{g}}$. The orbits of ${\rm Aut}_{c}(\mathfrak{g})$ are
given by the (topologically) connected components of each orbit of ${\rm
Aut}(\mathfrak{g})$. The trivial orbits given by $0\in\Lambda^{2}\mathfrak{g}$
are not considered. Symbol $A_{\pm}$ stands for two orbits, one with $+$ and
other with $-$. In these cases, $\mathbb{R}_{\pm}$ stands for $\mathbb{R}_{+}$
for the first orbit and $\mathbb{R}_{-}$ for the second. If an orbit of
Aut$(\mathfrak{g})$ in $\Lambda^{2}\mathfrak{g}$ belongs to
$\mathcal{Y}_{\mathfrak{g}}$ only for a certain set of parameters, each family
of possible values of the parameters is indicated in a subindex, first, or as
a superindex, if a second family of parameters is available. A star $(*)$ is
used to denote $r$-matrices that are not solutions to the CYBE.

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|  | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{23}$ | $e_{24}$ | $e_{34}$
---|---|---|---|---|---|---|---
$\mathfrak{s}_{1}$ | $e_{1}$ | 0 | 0 | 0 | 0 | 0 | 0
$e_{2}$ | 0 | 0 | 0 | 0 | $e_{12}$ | $e_{13}$
$e_{3}$ | 0 | 0 | $-e_{13}$ | 0 | $-e_{23}$ | 0
$e_{4}$ | 0 | $e_{13}$ | 0 | $e_{13}+e_{23}$ | $e_{14}$ | $e_{34}$
$\mathfrak{s}_{2}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $e_{12}$ | $e_{13}$
$e_{2}$ | 0 | 0 | $-e_{12}$ | 0 | $e_{12}$ | $e_{13}+e_{23}$
$e_{3}$ | 0 | 0 | $-e_{12}-e_{13}$ | 0 | $-e_{23}$ | $e_{23}$
$e_{4}$ | $2e_{12}$ | $e_{12}+2e_{13}$ | $e_{14}$ | $e_{13}+2e_{23}$ | $e_{14}+e_{24}$ | $e_{24}+e_{34}$
$\mathfrak{s}_{3}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $e_{12}$ | $e_{13}$
$e_{2}$ | 0 | 0 | $-\alpha e_{12}$ | 0 | 0 | $\alpha e_{23}$
$e_{3}$ | 0 | 0 | $-\beta e_{13}$ | 0 | $-\beta e_{23}$ | 0
$e_{4}$ | $(1+\alpha)e_{12}$ | $(1+\beta)e_{13}$ | $e_{14}$ | $(\alpha+\beta)e_{23}$ | $\alpha e_{24}$ | $\beta e_{34}$
$\mathfrak{s}_{4}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $e_{12}$ | $e_{13}$
$e_{2}$ | 0 | 0 | $-e_{12}$ | 0 | $e_{12}$ | $e_{13}+e_{23}$
$e_{3}$ | 0 | 0 | $-\alpha e_{13}$ | 0 | $-\alpha e_{23}$ | 0
$e_{4}$ | $2e_{12}$ | $(1+\alpha)e_{13}$ | $e_{14}$ | $e_{13}+(1+\alpha)e_{23}$ | $e_{14}+e_{24}$ | $\alpha e_{34}$
$\mathfrak{s}_{5}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $\alpha e_{12}$ | $\alpha e_{13}$
$e_{2}$ | 0 | 0 | $e_{13}-\beta e_{12}$ | 0 | $e_{23}$ | $\beta e_{23}$
$e_{3}$ | 0 | 0 | $-e_{12}-\beta e_{13}$ | 0 | $-\beta e_{23}$ | $e_{23}$
$e_{4}$ | $(\alpha+\beta)e_{12}-e_{13}$ | $e_{12}+(\alpha+\beta)e_{13}$ | $\alpha e_{14}$ | $2\beta e_{23}$ | $\beta e_{24}-e_{34}$ | $e_{24}+\beta e_{34}$
$\mathfrak{s}_{6}$ | $e_{1}$ | 0 | 0 | 0 | 0 | 0 | 0
$e_{2}$ | 0 | 0 | $-e_{12}$ | $-e_{12}$ | 0 | $e_{14}+e_{23}$
$e_{3}$ | 0 | 0 | $e_{13}$ | $-e_{13}$ | $-e_{14}+e_{23}$ | 0
$e_{4}$ | $e_{12}$ | $-e_{13}$ | 0 | 0 | $e_{24}$ | $-e_{34}$
$\mathfrak{s}_{7}$ | $e_{1}$ | 0 | 0 | 0 | 0 | 0 | 0
$e_{2}$ | 0 | 0 | $e_{13}$ | $-e_{12}$ | $e_{23}$ | $e_{14}$
$e_{3}$ | 0 | 0 | $-e_{12}$ | $-e_{13}$ | $-e_{14}$ | $e_{23}$
$e_{4}$ | $-e_{13}$ | $e_{12}$ | 0 | 0 | $-e_{34}$ | $e_{24}$
$\mathfrak{s}_{8}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $(1+\alpha)e_{12}$ | $(1+\alpha)e_{13}$
$e_{2}$ | 0 | 0 | $-e_{12}$ | $-e_{12}$ | 0 | $e_{14}+e_{23}$
$e_{3}$ | 0 | 0 | $-\alpha e_{13}$ | $-e_{13}$ | $-e_{14}-\alpha e_{23}$ | 0
$e_{4}$ | $(2+\alpha)e_{12}$ | $(1+2\alpha)e_{13}$ | $(1+\alpha)e_{14}$ | $(1+\alpha)e_{23}$ | $e_{24}$ | $\alpha e_{34}$
$\mathfrak{s}_{9}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $2\alpha e_{12}$ | $2\alpha e_{13}$
$e_{2}$ | 0 | 0 | $e_{13}-\alpha e_{12}$ | $-e_{12}$ | $e_{23}$ | $e_{14}+\alpha e_{23}$
$e_{3}$ | 0 | 0 | $-e_{12}-\alpha e_{13}$ | $-e_{13}$ | $-e_{14}-\alpha e_{23}$ | $e_{23}$
$e_{4}$ | $3\alpha e_{12}-e_{13}$ | $3\alpha e_{13}+e_{12}$ | $2\alpha e_{14}$ | $2\alpha e_{23}$ | $\alpha e_{24}-e_{34}$ | $e_{24}+\alpha e_{34}$
$\mathfrak{s}_{10}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $2e_{12}$ | $2e_{13}$
$e_{2}$ | 0 | 0 | $-e_{12}$ | $-e_{12}$ | 0 | $e_{14}+e_{23}$
$e_{3}$ | 0 | 0 | $-e_{12}-e_{13}$ | $-e_{13}$ | $-e_{14}-e_{23}$ | $e_{23}$
$e_{4}$ | $3e_{12}$ | $e_{12}+3e_{13}$ | $2e_{14}$ | $2e_{23}$ | $e_{24}$ | $e_{24}+e_{34}$
$\mathfrak{s}_{11}$ | $e_{1}$ | 0 | 0 | 0 | 0 | $e_{12}$ | $e_{13}$
$e_{2}$ | 0 | 0 | $-e_{12}$ | $-e_{12}$ | 0 | $e_{23}+e_{14}$
$e_{3}$ | 0 | 0 | 0 | $-e_{13}$ | $-e_{14}$ | 0
$e_{4}$ | $2e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{23}$ | $e_{24}$ | 0
$\mathfrak{s}_{12}$ | $e_{1}$ | 0 | 0 | $e_{12}$ | $e_{12}$ | 0 | $-e_{14}-e_{23}$
$e_{2}$ | 0 | $-e_{12}$ | 0 | 0 | $e_{12}$ | $-e_{24}+e_{13}$
$e_{3}$ | $2e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{23}$ | $e_{24}$ | 0
$e_{4}$ | 0 | $-e_{23}$ | $-e_{24}$ | $e_{13}$ | $e_{14}$ | 0
$\mathfrak{n}_{1}$ | $e_{1}$ | 0 | 0 | 0 | 0 | 0 | 0
$e_{2}$ | 0 | 0 | 0 | 0 | $-e_{12}$ | $-e_{13}$
$e_{3}$ | 0 | 0 | $e_{12}$ | 0 | 0 | $-e_{23}$
$e_{4}$ | 0 | $-e_{12}$ | 0 | $-e_{13}$ | $-e_{14}$ | $-e_{24}$
Table 3: Schouten brackets between basis elements of $\mathfrak{g}$ and
$\Lambda^{2}\mathfrak{g}$ for real four-dimensional indecomposable Lie
algebras.

|  | $e_{12}$ | $e_{13}$ | $e_{14}$ | $e_{23}$ | $e_{24}$ | $e_{34}$
---|---|---|---|---|---|---|---
$\mathfrak{s}_{1}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | 0
$e_{13}$ |  | 0 | 0 | 0 | $-e_{123}$ | 0
$e_{14}$ |  |  | 0 | $e_{123}$ | 0 | $e_{134}$
$e_{23}$ |  |  |  | 0 | $-e_{123}$ | 0
$e_{24}$ |  |  |  |  | $-2e_{124}$ | $-e_{134}+e_{234}$
$e_{34}$ |  |  |  |  |  | 0
$\mathfrak{s}_{2}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $2e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-2e_{123}$ | $e_{123}$
$e_{14}$ |  |  | 0 | $2e_{123}$ | 0 | $e_{124}$
$e_{23}$ |  |  |  | 0 | $-e_{123}$ | 0
$e_{24}$ |  |  |  |  | $-2e_{124}$ | $-e_{134}$
$e_{34}$ |  |  |  |  |  | $-2e_{234}$
$\mathfrak{s}_{3}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $(1+\alpha)e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-(1+\beta)e_{123}$ | 0
$e_{14}$ |  |  | 0 | $(\alpha+\beta)e_{123}$ | $(\alpha-1)e_{124}$ | $(\beta-1)e_{134}$
$e_{23}$ |  |  |  | 0 | 0 | 0
$e_{24}$ |  |  |  |  | 0 | $(\beta-\alpha)e_{234}$
$e_{34}$ |  |  |  |  |  | 0
$\mathfrak{s}_{4}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $2e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-(1+\alpha)e_{123}$ | 0
$e_{14}$ |  |  | 0 | $(1+\alpha)e_{123}$ | 0 | $(\alpha-1)e_{134}$
$e_{23}$ |  |  |  | 0 | $-e_{123}$ | 0
$e_{24}$ |  |  |  |  | $-2e_{124}$ |  $(\alpha-1)e_{234}\\\ -e_{134}$
$e_{34}$ |  |  |  |  |  | 0
$\mathfrak{s}_{5}$ | $e_{12}$ | 0 | 0 | 0 | 0 | $e_{123}$ | $(\alpha+\beta)e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-(\alpha+\beta)e_{123}$ | $e_{123}$
$e_{14}$ |  |  | 0 | $2\beta e_{123}$ |  $(\beta-\alpha)e_{124}\\\ -e_{134}$ |  $(\beta-\alpha)e_{134}\\\ +e_{124}$
$e_{23}$ |  |  |  | 0 | 0 | 0
$e_{24}$ |  |  |  |  | $-2e_{234}$ | 0
$e_{34}$ |  |  |  |  |  | $-2e_{234}$
$\mathfrak{s}_{6}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $e_{123}$ | 0
$e_{14}$ |  |  | 0 | 0 | $e_{124}$ | $-e_{134}$
$e_{23}$ |  |  |  | $2e_{123}$ | $e_{124}$ | $e_{134}$
$e_{24}$ |  |  |  |  | 0 | $-2e_{234}$
$e_{34}$ |  |  |  |  |  | 0

| $e_{12}$ | 0 | 0 | 0 | 0 | $e_{123}$ | 0
---|---|---|---|---|---|---|---
| $e_{13}$ |  | 0 | 0 | 0 | 0 | $e_{123}$
$\mathfrak{s}_{7}$ | $e_{14}$ |  |  | 0 | 0 | $-e_{134}$ | $e_{124}$
| $e_{23}$ |  |  |  | $2e_{123}$ | $e_{124}$ | $e_{134}$
| $e_{24}$ |  |  |  |  | $-2e_{234}$ | 0
| $e_{34}$ |  |  |  |  |  | $-2e_{234}$
$\mathfrak{s}_{8}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $(2+\alpha)e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-(1+2\alpha)e_{123}$ | 0
$e_{14}$ |  |  | 0 | $(1+\alpha)e_{123}$ | $-\alpha e_{124}$ | $-e_{134}$
$e_{23}$ |  |  |  | $2e_{123}$ | $e_{124}$ | $e_{134}$
$e_{24}$ |  |  |  |  | 0 | $(\alpha-1)e_{234}$
$e_{34}$ |  |  |  |  |  | 0
$\mathfrak{s}_{9}$ | $e_{12}$ | 0 | 0 | 0 | 0 | $e_{123}$ | $3\alpha e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-3\alpha e_{123}$ | $e_{123}$
$e_{14}$ |  |  | 0 | $2\alpha e_{123}$ | $-\alpha e_{124}-e_{134}$ | $e_{124}-\alpha e_{134}$
$e_{23}$ |  |  |  | $2e_{123}$ | $e_{124}$ | $e_{134}$
$e_{24}$ |  |  |  |  | $-2e_{234}$ | 0
$e_{34}$ |  |  |  |  |  | $-2e_{234}$
$\mathfrak{s}_{10}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $3e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-3e_{123}$ | $e_{123}$
$e_{14}$ |  |  | 0 | $2e_{123}$ | $-e_{124}$ | $e_{124}-e_{134}$
$e_{23}$ |  |  |  | $2e_{123}$ | $e_{124}$ | $e_{134}$
$e_{24}$ |  |  |  |  | 0 | 0
$e_{34}$ |  |  |  |  |  | $-2e_{234}$
$\mathfrak{s}_{11}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $2e_{123}$
$e_{13}$ |  | 0 | 0 | 0 | $-e_{123}$ | 0
$e_{14}$ |  |  | 0 | $e_{123}$ | 0 | $-e_{134}$
$e_{23}$ |  |  |  | $2e_{123}$ | $e_{124}$ | $e_{134}$
$e_{24}$ |  |  |  |  | 0 | $-e_{234}$
$e_{34}$ |  |  |  |  |  | 0
$\mathfrak{s}_{12}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | $-2e_{124}$
$e_{13}$ |  | 0 | $-e_{123}$ | 0 | $e_{124}$ | $-e_{134}$
$e_{14}$ |  |  | $-2e_{124}$ | $-e_{124}$ | 0 | $e_{234}$
$e_{23}$ |  |  |  | 0 | $-e_{123}$ | $-e_{234}$
$e_{24}$ |  |  |  |  | $-2e_{124}$ | $-e_{134}$
$e_{34}$ |  |  |  |  |  | 0
$\mathfrak{n}_{1}$ | $e_{12}$ | 0 | 0 | 0 | 0 | 0 | 0
$e_{13}$ |  | 0 | 0 | 0 | 0 | $-e_{123}$
$e_{14}$ |  |  | 0 | 0 | 0 | $-e_{124}$
$e_{23}$ |  |  |  | 0 | $e_{123}$ | 0
$e_{24}$ |  |  |  |  | $2e_{124}$ | $e_{134}$
$e_{34}$ |  |  |  |  |  | $2e_{234}$

Table 4: Schouten brackets between basis elements of $\Lambda^{2}\mathfrak{g}$ for four-dimensional indecomposable Lie algebras. |  | $e_{123}$ | $e_{124}$ | $e_{134}$ | $e_{234}$
---|---|---|---|---|---
$\mathfrak{s}_{1}$ | $e_{1}$ | 0 | 0 | 0 | 0
$e_{2}$ | 0 | 0 | 0 | $-e_{123}$
$e_{3}$ | 0 | $-e_{123}$ | 0 | 0
$e_{4}$ | $e_{123}$ | $0$ | $e_{134}$ | $e_{134}+e_{234}$
$\mathfrak{s}_{2}$ | $e_{1}$ | 0 | 0 | 0 | $-e_{123}$
$e_{2}$ | 0 | 0 | $e_{123}$ | $-e_{123}$
$e_{3}$ | 0 | $-e_{123}$ | $e_{123}$ | 0
$e_{4}$ | $3e_{123}$ | $2e_{124}$ | $e_{124}+2e_{134}$ | $e_{134}+2e_{234}$
$\mathfrak{s}_{3}$ | $e_{1}$ | 0 | 0 | 0 | $-e_{123}$
$e_{2}$ | 0 | 0 | $\alpha e_{123}$ | 0
$e_{3}$ | 0 | $-\beta e_{123}$ | 0 | 0
$e_{4}$ | $(1+\alpha+\beta)e_{123}$ | $(1+\alpha)e_{124}$ | $(1+\beta)e_{134}$ | $(\alpha+\beta)e_{234}$
$\mathfrak{s}_{4}$ | $e_{1}$ | 0 | 0 | 0 | $e_{123}$
$e_{2}$ | 0 | 0 | $e_{123}$ | $-e_{123}$
$e_{3}$ | 0 | $-\alpha e_{123}$ | 0 | 0
$e_{4}$ | $(2+\alpha)e_{123}$ | $2e_{124}$ | $(1+\alpha)e_{134}$ | $e_{134}+(1+\alpha)e_{234}$
$\mathfrak{s}_{5}$ | $e_{1}$ | 0 | 0 | 0 | $-\alpha e_{123}$
$e_{2}$ | 0 | $e_{123}$ | $\beta e_{123}$ | 0
$e_{3}$ | 0 | $-\beta e_{123}$ | $e_{123}$ | 0
$e_{4}$ | $(\alpha+2\beta)e_{123}$ | $(\alpha+\beta)e_{124}-e_{134}$ | $e_{124}+(\alpha+\beta)e_{134}$ | $2e_{234}$
$\mathfrak{s}_{6}$ | $e_{1}$ | 0 | 0 | 0 | 0
$e_{2}$ | 0 | 0 | $e_{123}$ | $-e_{124}$
$e_{3}$ | 0 | $e_{123}$ | 0 | $-e_{134}$
$e_{4}$ | 0 | $e_{124}$ | $-e_{134}$ | 0
$\mathfrak{s}_{7}$ | $e_{1}$ | 0 | 0 | 0 | 0
$e_{2}$ | 0 | $e_{123}$ | 0 | $-e_{124}$
$e_{3}$ | 0 | 0 | $e_{123}$ | $-e_{134}$
$e_{4}$ | 0 | $-e_{134}$ | $e_{124}$ | 0
$\mathfrak{s}_{8}$ | $e_{1}$ | 0 | 0 | 0 | $-(1+\alpha)e_{123}$
$e_{2}$ | 0 | 0 | $e_{123}$ | $-e_{124}$
$e_{3}$ | 0 | $-\alpha e_{123}$ | 0 | $-e_{134}$
$e_{4}$ | $2(1+\alpha)e_{123}$ | $(2+\alpha)e_{124}$ | $(1+2\alpha)e_{134}$ | $(1+\alpha)e_{234}$
$\mathfrak{s}_{9}$ | $e_{1}$ | 0 | 0 | 0 | $-2\alpha e_{123}$
$e_{2}$ | 0 | $e_{123}$ | $\alpha e_{123}$ | $-e_{124}$
$e_{3}$ | 0 | $-\alpha e_{123}$ | $e_{123}$ | $-e_{134}$
$e_{4}$ | $4\alpha e_{123}$ | $3\alpha e_{124}-e_{134}$ | $3\alpha e_{134}+e_{124}$ | $2\alpha e_{234}$
$\mathfrak{s}_{10}$ | $e_{1}$ | 0 | 0 | 0 | $-2e_{123}$
$e_{2}$ | 0 | 0 | $e_{123}$ | $-e_{124}$
$e_{3}$ | 0 | $-e_{123}$ | $e_{123}$ | $-e_{134}$
$e_{4}$ | $4e_{123}$ | $3e_{124}$ | $e_{124}+3e_{134}$ | $2e_{234}$
$\mathfrak{s}_{11}$ | $e_{1}$ | 0 | 0 | 0 | $-e_{123}$
$e_{2}$ | 0 | 0 | $e_{123}$ | $-e_{124}$
$e_{3}$ | 0 | 0 | 0 | $-e_{134}$
$e_{4}$ | $2e_{123}$ | $2e_{124}$ | $e_{134}$ | $e_{234}$
$\mathfrak{s}_{12}$ | $e_{1}$ | 0 | 0 | $-e_{123}$ | $-e_{124}$
$e_{2}$ | 0 | 0 | $-e_{124}$ | $-e_{123}$
$e_{3}$ | $2e_{123}$ | $2e_{124}$ | $e_{134}$ | $e_{234}$
$e_{4}$ | 0 | 0 | $-e_{234}$ | $e_{134}$
$\mathfrak{n}_{1}$ | $e_{1}$ | 0 | 0 | 0 | 0
$e_{2}$ | 0 | 0 | 0 | $e_{123}$
$e_{3}$ | 0 | 0 | $-e_{123}$ | 0
$e_{4}$ | 0 | 0 | $-e_{124}$ | $-e_{134}$
Table 5: Schouten brackets between basis elements of $\mathfrak{g}$ and
$\Lambda^{3}\mathfrak{g}$ for real four-dimensional indecomposable Lie
algebras.