  
  [1X3 A description of the Lie algebras that are contained in the package[0X
  
  
  [1X3.1 Description of the non-solvable Lie algebras[0X
  
  In  this  section  we  list  the  non-solvable Lie algebras contained in the
  package.  Our  notation follows [Str], 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 $F<x_1>$
  
  
  [1X3.2 Dimension 3[0X
  
  There  are  no  non-solvable  Lie  algebras  with  dimension 1 or 2. Over an
  arbitrary finite field [3XF[0m, there is just one isomorphism type of non-solvable
  Lie algebras:
  
  (1)   If [3Xchar F=2[0m then the algebra is $W(1;\underline 2)^{(1)}$.
  
  (2)   If [3Xchar F>2[0m then the algebra is $\mbox{sl}(2,F)$.
  
  See Theorem 3.2 of [Str] for details.
  
  
  [1X3.3 Dimension 4[0X
  
  Over  a finite field [3XF[0m of characteristic 2 there are two isomorphism classes
  of  non-solvable  Lie algebras with dimension 4, while over a finite field [3XF[0m
  of  odd characteristic the number of isomorphism classes is one (see Theorem
  4.1 of [Str]). The classes are as follows:
  
  (1)   characteristic    2:    $W(1;\underline    2)$   and   $W(1;\underline
        2)^{(1)}\oplus F$.
  
  (2)   odd characteristic: $\mbox{gl}(2,F)$.
  
  
  [1X3.4 Dimension 5[0X
  
  
  [1X3.4-1 Characteristic 2[0X
  
  Over a finite field [3XF[0m 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 [Str] for details.
  
  
  [1X3.4-2 Odd characteristic[0X
  
  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 [Str]). 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:
        $W(1;\underline 1)$.
  
  
  [1X3.5 Dimension 6[0X
  
  
  [1X3.5-1 Characteristic 2[0X
  
  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  [Str]  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 [Str].
  
  
  [1X3.5-2 General odd characteristic[0X
  
  If   the  characteristic  of  the  field  is  odd,  then  the  6-dimensional
  non-solvable  Lie algebras are described by Theorems 5.2--5.4 of [Str]. 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.
  
  
  [1X3.5-3 Characteristic 3[0X
  
  If  the  characteristic  of the field $F$ is 3, then, besides the classes in
  Section [14X3.5-2[0m, 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  $\xi$  such  that $T^3+T^2-\xi$ has a root in $F$; see
        Proposition 3.5 of [Str] 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 [Str]. We note
        that Proposition 4.5 of [Str] 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 [Str].
  
  We note here that [Str] 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).
  
  
  [1X3.5-4 Characteristic 5[0X
  
  If  the  characteristic  of the field $F$ is 5, then, besides the classes in
  Section [14X3.5-2[0m, we also obtain the following isomorphism classes.
  
  (1)   $W(1;\underline 1)\oplus F$.
  
  (2)   The    non-split    central   extension   $0\rightarrow   F\rightarrow
        L\rightarrow W(1;\underline 1)\rightarrow 0$.
  
  
  [1X3.6 Description of the simple Lie algebras[0X
  
  If  [3XF[0m  is  a  finite  field, then, up to isomorphism, there is precisely one
  simple Lie algebra with dimension 3, and another one with dimension 6; these
  can    be    accessed    by   calling   [3XNonSolvableLieAlgebra(F,[3,1])[0m   and
  [3XNonSolvableLieAlgebra(F,[6,2])[0m  (see [3XNonSolvableLieAlgebra[0m for the details).
  Over  a field of characteristic 5, there is an additional simple Lie algebra
  with  dimension 5, namely [3XNonSolvableLieAlgebra(F,[5,3])[0m. These are the only
  isomorphism  types of simple Lie algebras over finite fields up to dimension
  6.
  
  In  addition  to  the  algebras  above  the  package contains the simple Lie
  algebras  of  dimension  between 7 and 9 over [3XGF(2)[0m. These Lie algebras were
  determined by [Vau06] and can be described as follows.
  
  There  are two isomorphism classes of 7-dimensional Lie algebras over [3XGF(2)[0m.
  In a basis b1,...,b7 the non-trivial products in the first algebra are
  
  [b1,b2]=b3, [b1,b3]=b4, [b1,b4]=b5, [b1,b5]=b6
  [b1,b6]=b7, [b1,b7]=b1, [b2,b7]=b2, [b3,b6]=b2, 
  [b4,b5]=b2, [b4,b6]=b3, [b4,b7]=b4, [b6,b7]=b6;
  
  and those in the second are
  
  [b1,b2]=b3, [b1,b3]=b1+b4, [b1,b4]=b5, [b1,b5]=b6, 
  [b1,b6]=b7, [b2,b3]=b2, [b2,b5]=b2+b4, [b2,b6]=b5, 
  [b2,b7]=b1+b4, [b3,b4]=b2+b4, [b3,b5]=b3, [b3,b6]=b1+b4+b6, 
  [b3,b7]=b5, [b4,b7]=b6, [b5,b6]=b6, [b5,b7]=b7.
  
  Over  [3XGF(2)[0m  there  are  two  isomorphism  types of simple Lie algebras with
  dimension  8.  In the basis b1,...,b8 the non-trivial products for the first
  one are
  
  [b1,b3]=b5, [b1,b4]=b6, [b1,b7]=b2, [b1,b8]=b1, [b2,b3]=b7, [b2,b4]=b5+b8, 
  [b2,b5]=b2, [b2,b6]=b1, [b2,b8]=b2, [b3,b6]=b4, [b3,b8]=b3, [b4,b5]=b4, 
  [b4,b7]=b3, [b4,b8]=b4, [b5,b6]=b6, [b5,b7]=b7, [b6,b7]=b8;
  
  and for the second one they are
  
  [b1,b2]=b3, [b1,b3]=b2+b5, [b1,b4]=b6, [b1,b5]=b2, [b1,b6]=b1+b4+b8, 
  [b1,b8]=b4, [b2,b3]=b4, [b2,b4]=b1, [b2,b5]=b6, [b2,b6]=b2+b7, 
  [b2,b7]=b2+b5, [b3,b4]=b2+b7, [b3,b5]=b1+b4+b8, [b3,b6]=b1, [b3,b7]=b2+b3, 
  [b3,b8]=b1, [b4,b5]=b3, [b4,b6]=b2+b4, [b4,b7]=b1+b4+b8, [b4,b8]=b3, 
  [b5,b6]=b1+b2+b5, [b5,b7]=b3, [b5,b8]=b2+b7, [b6,b7]=b4+b6, [b6,b8]=b2+b5, 
  [b7,b8]=b6.
  
  The  non-trivial products for the unique simple Lie algebra with dimension 9
  over [3XGF(2)[0m are as follows:
  
  [b1,b2]=b3, [b1,b3]=b5, [b1,b5]=b6, [b1,b6]=b7, [b1,b7]=b6+b9, 
  [b1,b9]=b2, [b2,b3]=b4, [b2,b4]=b6, [b2,b6]=b8, [b2,b8]=b6+b9, 
  [b2,b9]=b1, [b3,b4]=b7, [b3,b5]=b8, [b3,b7]=b1+b8, [b3,b8]=b2+b7, 
  [b4,b5]=b6+b9, [b4,b6]=b2+b7, [b4,b7]=b3+b6+b9, [b4,b9]=b5, 
  [b5,b6]=b1+b8, [b5,b8]=b3+b6+b9, [b5,b9]=b4, [b6,b7]=b1+b4+b8, 
  [b6,b8]=b2+b5+b7, [b7,b8]=b3+b9, [b7,b9]=b8, [b8,b9]=b7.
  
  
  [1X3.7 Description of the solvable and nilpotent Lie algebras[0X
  
  In  this  section  we  list  the  multiplication tables of the nilpotent and
  solvable  Lie  algebras  contained  in  the package. Some parametric classes
  contain isomorphic Lie algebras, for different values of the parameters. For
  exact  descriptions  of  these  isomorphisms  we refer to [dG05], [dG07]. In
  dimension 2 there are just two classes of solvable Lie algebras:
  
  --    L_2^1: The Abelian Lie algebra.
  
  --    L_2^2: [x_2,x_1]=x_1.
  
  We have the following solvable Lie algebras of dimension 3:
  
  --    L_3^1 The Abelian Lie algebra.
  
  --    L_3^2 [x_3,x_1]=x_1, [x_3,x_2]=x_2.
  
  --    L_3^3(a) [x_3,x_1]=x_2, [x_3,x_2]=ax_1+x_2.
  
  --    L_3^4(a) [x_3,x_1]=x_2, [x_3,x_2]=ax_1.
  
  And the following solvable Lie algebras of dimension 4:
  
  --    L_4^1 The Abelian Lie algebra.
  
  --    L_4^2 [x_4,x_1]=x_1, [x_4,x_2]=x_2, [x_4,x_3]=x_3.
  
  --    L_4^3(a) [x_4,x_1]=x_1, [x_4,x_2]=x_3, [x_4,x_3]=-ax_2 +(a+1)x_3 .
  
  --    L_4^4 [x_4,x_2]=x_3, [x_4,x_3]= x_3 .
  
  --    L_4^5 [x_4,x_2]=x_3 .
  
  --    L_4^6(a,b) [x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2+x_3 .
  
  --    L_4^7(a,b) [x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2.
  
  --    L_4^8 [x_1,x_2]=x_2, [x_3,x_4]=x_4 .
  
  --    L_4^9(a)   [x_4,x_1]   =   x_1+ax_2,   [x_4,x_2]=x_1,   [x_3,x_1]=x_1,
        [x_3,x_2]=x_2 .
  
  --    L_4^10(a)    [x_4,x_1]    =    x_2,   [x_4,x_2]=ax_1,   [x_3,x_1]=x_1,
        [x_3,x_2]=x_2 Condition on F: the characteristic of F is 2.
  
  --    L_4^11(a,b)  [x_4,x_1]  =  x_1,  [x_4,x_2] = bx_2, [x_4,x_3]=(1+b)x_3,
        [x_3,x_1]=x_2, [x_3,x_2]=ax_1. Condition on F: the characteristic of F
        is 2.
  
  --    L_4^12 [x_4,x_1] = x_1, [x_4,x_2]=2x_2, [x_4,x_3] = x_3, [x_3,x_1]=x_2
        .
  
  --    L_4^13(a)  [x_4,x_1]  =  x_1+ax_3,  [x_4,x_2]=x_2,  [x_4,x_3]  =  x_1,
        [x_3,x_1]=x_2 .
  
  --    L_4^14(a) [x_4,x_1] = ax_3, [x_4,x_3]=x_1, [x_3,x_1]=x_2 .
  
  Nilpotent of dimension 5:
  
  --    N_5,1 Abelian.
  
  --    N_5,2 [x_1,x_2]=x_3 .
  
  --    N_5,3 [x_1,x_2]=x_3, [x_1,x_3]=x_4 .
  
  --    N_5,4 [x_1,x_2] = x_5, [x_3,x_4]=x_5 .
  
  --    N_5,5 [x_1,x_2]=x_3, [x_1,x_3]= x_5, [x_2,x_4] = x_5 .
  
  --    N_5,6 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5 .
  
  --    N_5,7 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5 .
  
  --    N_5,8 [x_1,x_2]=x_4, [x_1,x_3]=x_5 .
  
  --    N_5,9 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5 .
  
  We get nine 6-dimensional nilpotent Lie algebras denoted N_6,k for k=1,...,9
  that  are  the  direct  sum  of  N_5,k  and  a  1-dimensional abelian ideal.
  Subsequently we get the following Lie algebras.
  
  --    N_6,10 [x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_4,x_5]=x_6.
  
  --    N_6,11  [x_1,x_2]=x_3,  [x_1,x_3]=x_4,  [x_1,x_4]=x_6,  [x_2,x_3]=x_6,
        [x_2,x_5]=x_6 .
  
  --    N_6,12 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_5]=x_6 .
  
  --    N_6,13  [x_1,x_2]=x_3,  [x_1,x_3]=x_5,  [x_2,x_4]=x_5,  [x_1,x_5]=x_6,
        [x_3,x_4]=x_6 .
  
  --    N_6,14  [x_1,x_2]=x_3,  [x_1,x_3]=x_4,  [x_1,x_4]=x_5,  [x_2,x_3]=x_5,
        [x_2,x_5]=x_6,[x_3,x_4]=-x_6 .
  
  --    N_6,15  [x_1,x_2]=x_3,  [x_1,x_3]=x_4,  [x_1,x_4]=x_5,  [x_2,x_3]=x_5,
        [x_1,x_5]=x_6,[x_2,x_4]=x_6 .
  
  --    N_6,16  [x_1,x_2]=x_3,  [x_1,x_3]=x_4,  [x_1,x_4]=x_5,  [x_2,x_5]=x_6,
        [x_3,x_4]=-x_6 .
  
  --    N_6,17  [x_1,x_2]=x_3,  [x_1,x_3]=x_4,  [x_1,x_4]=x_5,  [x_1,x_5]=x_6,
        [x_2,x_3]= x_6 .
  
  --    N_6,18 [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6 .
  
  --    N_6,19(a) [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_4]=x_6, [x_3,x_5]=a x_6
        .
  
  --    N_6,20 [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_6 .
  
  --    N_6,21(a)  [x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5, [x_1,x_4]=x_6,
        [x_2,x_5]= a x_6 .
  
  --    N_6,22(a)    [x_1,x_2]=x_5,    [x_1,x_3]=x_6,    [x_2,x_4]=   a   x_6,
        [x_3,x_4]=x_5 .
  
  --    N_6,23 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6, [x_2,x_4]= x_5 .
  
  --    N_6,24(a)     [x_1,x_2]=x_3,     [x_1,x_3]=x_5,    [x_1,x_4]=a    x_6,
        [x_2,x_3]=x_6, [x_2,x_4]= x_5 .
  
  --    N_6,25 [x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6 .
  
  --    N_6,26 [x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6 .
  
