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### B. Description of the non-solvable Lie algebras

In this appendix we list the non-solvable Lie algebras contained in the package. Our notation follows [S], where a more detailed description can also be found. In particular if $L$ is a Lie algebra over $F$ then $C(L)$ denotes the center of $L$. Further, if $x_1,\ldots,x_k$ are elements of $L$, then $F<x_1,\ldots,x_k>$ denotes the linear subspace generated by $x_1,\ldots,x_k$, and we also write $Fx_1$ for $F<x_1>$

#### B.1 Dimension 3

There are no non-solvable Lie algebras with dimension 1 or 2. Over an arbitrary finite field F, there is just one isomorphism type of non-solvable Lie algebras:

1. If char F=2 then the algebra is $W(1;\underline 2)^{(1)}$.

2. If char F>2 then the algebra is $\mbox{sl}(2,F)$.

See Theorem~3.2 of [S] for details.

#### B.2 Dimension 4

Over a finite field F of characteristic 2 there are two isomorphism classes of non-solvable Lie algebras with dimension 4, while over a finite field F of odd characteristic the number of isomorphism classes is one (see Theorem~4.1 of [S]). The classes are as follows:

1. characteristic 2: $W(1;\underline 2)$ and $W(1;\underline 2)^{(1)}\oplus F$.

2. odd characteristic: $\mbox{gl}(2,F)$.

#### B.3 Dimension 5

##### B.3-1 Characteristic 2

Over a finite field F of characteristic 2 there are 5 isomorphism classes of non-solvable Lie algebras with dimension 5:

1. $\mbox{Der}(W(1;\underline 2)^{(1)})$;

2. $W(1;\underline 2)\ltimes Fu$ where $[W(1;\underline 2)^{(1)},u]=0$, $[x^{(3)}\partial,u]=\delta u$ and $\delta\in\{0,1\}$ (two algebras);

3. $W(1;\underline 2)^{(1)}\oplus(F\left< h,u\right>)$, $[h,u]=\delta u$, where $\delta\in\{0,1\}$ (two algebras).

See Theorem 4.2 of [S] for details.

##### B.3-2 Odd characteristic

Over a field $F$of odd characteristic the number of isomorphism types of 5-dimensional non-solvable Lie algebras is $3$ if the characteristic is at least 7, and it is 4 otherwise (see Theorem 4.3 of [S]). The classes are as follows.

1. $\mbox{sl}(2,F)\oplus F<x,y>$, $[x,y]=\delta y$ where $\delta\in\{0,1\}$.

2. $\mbox{sl}(2,F)\ltimes V(1)$ where $V(1)$ is the irreducible 2-dimensional $\mbox{sl}(2,F)$=module.

3. If $\mbox{char }F=3$ then there is an additional algebra, namely the non-split extension $0\rightarrow V(1)\rightarrow L\rightarrow\mbox{sl}(2,F)\rightarrow 0$.

4. If $\mbox{char }F=5$ then there is an additional algebra: $W(1;\underline 1)$.

#### B.4 Dimension 6

##### B.4-1 Characteristic 2

Over a field $F$ of characteristic 2, the isomorphism classes of non-solvable Lie algebras are as follows.

1. $W(1;\underline 2)^{(1)}\oplus W(1;\underline 2)^{(1)}$.

2. $W(1;\underline 2)^{(1)}\otimes F_{q^2}$ where $F=F_q$.

3. $\mbox{Der}(W(1;\underline 2)^{(1)})\ltimes Fu$, $[W(1;\underline 2),u]=0$, $[\partial^2,u]=\delta u$ where $\delta=\{0,1\}$.

4. $W(1;\underline 2)\ltimes (F<h,u>)$, $[W(1;\underline 2)^{(1)},(F<h,u>]=0$, $[h,u]=\delta u$, and if $\delta=0$, then the action of $x^{(3)}\partial$ on $F<h,u>$ is given by one of the following matrices:

   \[ \left(\begin{array}{cc} 0 & 0\\ 0 & 0\end{array}\right),\ \left(\begin{array}{cc} 0 & 1\\ 0 & 0\end{array}\right),\ \left(\begin{array}{cc} 1 & 0\\ 0 & 1\end{array}\right),\ \left(\begin{array}{cc} 1 & 1\\ 0 & 1\end{array}\right),\ \left(\begin{array}{cc} 0 & \xi\\ 1 & 1\end{array}\right)\mbox{ where }\xi\in F^*. \]

5. the algebra is as in (4.), but $\delta=1$. Note that Theorem 5.1(3/b) of [S] lists two such algebras but they turn out to be isomorphic. We take the one with $[x^{(3)}\partial,h]=[x^{(3)}\partial,u]=0$.

6. $W(1;\underline 2)^{(1)}\oplus K$ where $K$ is a 3-dimensional solvable Lie algebra.

7. $W(1;\underline 2)^{(1)}\ltimes \mathcal O(1;\underline 2)/F$.

8. the non-split extension $0\rightarrow \mathcal O(1;\underline 2)/F\rightarrow L\rightarrow W(1;\underline 2)^{(1)}\rightarrow 0$.

See Theorem 5.1 of [S].

##### B.4-2 General odd characteristic

If the characteristic of the field is odd, then the 6-dimensional non-solvable Lie algebras are described by Theorem 5.2--5.4 of [S]. Over such a field $F$, let us define the following isomorphism classes of 6-dimensional non-solvable Lie algebras.

1. $\mbox{sl}(2,F)\oplus\mbox{sl}(2,F) $.

2. $\mbox{sl}(2,F_{q^2})$ where $F=F_q$;

3. $\mbox{sl}(2,F)\oplus K$ where $K$ is a solvable Lie algebra with dimension 3;

4. $\mbox{sl}(2,F)\ltimes (V(0)\oplus V(1))$ where $V(i)$ is the $(i+1)$-dimensional irreducible $\mbox{sl}(2,F)$-module;

5. $\mbox{sl}(2,F)\ltimes V(2)$ where $V(2)$ is the $3$-dimensional irreducible $\mbox{sl}(2,F)$-module;

6. $\mbox{sl}(2,F)\ltimes(V(1)\oplus C(L))\cong \mbox{sl}(2,F)\ltimes H$ where $H$ is the Heisenberg Lie algebra;

7. $\mbox{sl}(2,F)\ltimes K$ where $K=Fd\oplus K^{(1)}$, $K^{(1)}$ is 2-dimensional abelian, isomorphic, as an $\mbox{sl}(2,F)$-module, to $V(1)$, $[\mbox{sl}(2,F),d]=0$, and, for all $v\in K$, $[d,v]=v$;

If the characteristic of $F$ is at least 7, then these algebras form a complete and irredundant list of the isomorphism classes of the 6-dimensional non-solvable Lie algebras.

##### B.4-3 Characteristic 3

If the characteristic of the field $F$ is 3, then, besides the classes in Section B.4-2, we also obtain the following isomorphism classes.

1. $\mbox{sl}(2,F)\ltimes V(2,\chi)$ where $\chi$ is a 3-dimensional character of $\mbox{sl}(2,F)$. Each such character is described by a field element $a$ such that $T^3+T^2-a$ has a root in $F$; see Proposition 3.5 of [S] for more details.

2. $W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)$ where $\mathcal O(1;\underline 1)$ is considered as an abelian Lie algebra.

3. $W(1;\underline 1)\ltimes\mathcal O(1;\underline 1)^*$ where $\mathcal O(1;\underline 1)^*$ is the dual of $\mathcal O(1;\underline 1)$ and it is considered as an abelian Lie algebra.

4. One of the two 6-dimensional central extensions of the non-split extension $0\rightarrow V(1)\rightarrow L\rightarrow \mbox{sl}(2,F)\rightarrow 0$; see Proposition 4.5 of [S]. We note that Proposition 4.5 of [S] lists three such central extensions, but one of them is not a Lie algebra.

5. One of the two non-split extensions $0\rightarrow\mbox{rad } L\rightarrow L\rightarrow L/\mbox{rad } L\rightarrow 0$ with a 5-dimensional ideal; see Theorem~5.4 of [S].

We note here that [S] lists one more non-solvable Lie algebra over a field of characteristic 3, namely the one in Theorem~5.3(5). However, this algebra is isomorphic to the one in Theorem~5.3(4).

##### B.4-4 Characteristic 5

If the characteristic of the field $F$ is 5, then, besides the classes in Section~B.4-2, we also obtain the following isomorphism classes.

1. $W(1;\underline 1)\oplus F$.

2. The non-split central extension $0\rightarrow F\rightarrow L\rightarrow W(1;\underline 1)\rightarrow 0$.

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