  
  [1X3 [33X[0;0YA description of the Lie algebras that are contained in the package[133X[101X
  
  
  [1X3.1 [33X[0;0YDescription of the non-solvable Lie algebras[133X[101X
  
  [33X[0;0YIn  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 [23XL[123X is a Lie algebra over [23XF[123X then [23XC(L)[123X denotes
  the  center  of  [23XL[123X.  Further,  if  [23Xx_1,\ldots,x_k[123X  are  elements  of [23XL[123X, then
  [23XF<x_1,\ldots,x_k>[123X  denotes  the linear subspace generated by [23Xx_1,\ldots,x_k[123X,
  and we also write [23XFx_1[123X for [23XF<x_1>[123X[133X
  
  
  [1X3.2 [33X[0;0YDimension 3[133X[101X
  
  [33X[0;0YThere  are  no  non-solvable  Lie  algebras  with  dimension 1 or 2. Over an
  arbitrary finite field [3XF[103X, there is just one isomorphism type of non-solvable
  Lie algebras:[133X
  
  [31X1[131X   [33X[0;6YIf [3Xchar F=2[103X then the algebra is [23XW(1;\underline 2)^{(1)}[123X.[133X
  
  [31X2[131X   [33X[0;6YIf [3Xchar F>2[103X then the algebra is [23X\mbox{sl}(2,F)[123X.[133X
  
  [33X[0;0YSee Theorem 3.2 of [Str] for details.[133X
  
  
  [1X3.3 [33X[0;0YDimension 4[133X[101X
  
  [33X[0;0YOver  a finite field [3XF[103X of characteristic 2 there are two isomorphism classes
  of  non-solvable  Lie algebras with dimension 4, while over a finite field [3XF[103X
  of  odd characteristic the number of isomorphism classes is one (see Theorem
  4.1 of [Str]). The classes are as follows:[133X
  
  [31X1[131X   [33X[0;6Ycharacteristic  2: [23XW(1;\underline 2)[123X and [23XW(1;\underline 2)^{(1)}\oplus
        F[123X.[133X
  
  [31X2[131X   [33X[0;6Yodd characteristic: [23X\mbox{gl}(2,F)[123X.[133X
  
  
  [1X3.4 [33X[0;0YDimension 5[133X[101X
  
  
  [1X3.4-1 [33X[0;0YCharacteristic 2[133X[101X
  
  [33X[0;0YOver a finite field [3XF[103X of characteristic 2 there are 5 isomorphism classes of
  non-solvable Lie algebras with dimension 5:[133X
  
  [31X1[131X   [33X[0;6Y[23X\mbox{Der}(W(1;\underline 2)^{(1)})[123X;[133X
  
  [31X2[131X   [33X[0;6Y[23XW(1;\underline   2)\ltimes  Fu[123X  where  [23X[W(1;\underline  2)^{(1)},u]=0[123X,
        [23X[x^{(3)}\partial,u]=\delta u[123X and [23X\delta\in\{0,1\}[123X (two algebras);[133X
  
  [31X3[131X   [33X[0;6Y[23XW(1;\underline  2)^{(1)}\oplus(F\left<  h,u\right>)[123X,  [23X[h,u]=\delta  u[123X,
        where [23X\delta\in\{0,1\}[123X (two algebras).[133X
  
  [33X[0;0YSee Theorem 4.2 of [Str] for details.[133X
  
  
  [1X3.4-2 [33X[0;0YOdd characteristic[133X[101X
  
  [33X[0;0YOver  a  field  [23XF[123Xof  odd  characteristic  the number of isomorphism types of
  5-dimensional  non-solvable  Lie  algebras  is [23X3[123X if the characteristic is at
  least  7,  and it is 4 otherwise (see Theorem 4.3 of [Str]). The classes are
  as follows.[133X
  
  [31X1[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\oplus F<x,y>[123X, [23X[x,y]=\delta y[123X where [23X\delta\in\{0,1\}[123X.[133X
  
  [31X2[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\ltimes V(1)[123X where [23XV(1)[123X is the irreducible 2-dimensional
        [23X\mbox{sl}(2,F)[123X-module.[133X
  
  [31X3[131X   [33X[0;6YIf  [23X\mbox{char  }F=3[123X  then  there is an additional algebra, namely the
        non-split         extension        [23X0\rightarrow        V(1)\rightarrow
        L\rightarrow\mbox{sl}(2,F)\rightarrow 0[123X.[133X
  
  [31X4[131X   [33X[0;6YIf [23X\mbox{char }F=5[123X then there is an additional algebra: [23XW(1;\underline
        1)[123X.[133X
  
  
  [1X3.5 [33X[0;0YDimension 6[133X[101X
  
  
  [1X3.5-1 [33X[0;0YCharacteristic 2[133X[101X
  
  [33X[0;0YOver  a field [23XF[123X of characteristic 2, the isomorphism classes of non-solvable
  Lie algebras are as follows.[133X
  
  [31X1[131X   [33X[0;6Y[23XW(1;\underline 2)^{(1)}\oplus W(1;\underline 2)^{(1)}[123X.[133X
  
  [31X2[131X   [33X[0;6Y[23XW(1;\underline 2)^{(1)}\otimes F_{q^2}[123X where [23XF=F_q[123X.[133X
  
  [31X3[131X   [33X[0;6Y[23X\mbox{Der}(W(1;\underline    2)^{(1)})\ltimes    Fu[123X,   [23X[W(1;\underline
        2),u]=0[123X, [23X[\partial^2,u]=\delta u[123X where [23X\delta=\{0,1\}[123X.[133X
  
  [31X4[131X   [33X[0;6Y[23XW(1;\underline        2)\ltimes        (F<h,u>)[123X,       [23X[W(1;\underline
        2)^{(1)},(F<h,u>]=0[123X,  [23X[h,u]=\delta u[123X, and if [23X\delta=0[123X, then the action
        of  [23Xx^{(3)}\partial[123X  on  [23XF<h,u>[123X  is  given  by  one  of  the following
        matrices:[133X
  
  
  [24X      [33X[0;6Y\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^*.[133X
  
  [124X
  
  [31X5[131X   [33X[0;6Ythe algebra is as in (4.), but [23X\delta=1[123X. 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 [23X[x^{(3)}\partial,h]=[x^{(3)}\partial,u]=0[123X.[133X
  
  [31X6[131X   [33X[0;6Y[23XW(1;\underline  2)^{(1)}\oplus  K[123X  where [23XK[123X is a 3-dimensional solvable
        Lie algebra.[133X
  
  [31X7[131X   [33X[0;6Y[23XW(1;\underline 2)^{(1)}\ltimes \mathcal O(1;\underline 2)/F[123X.[133X
  
  [31X8[131X   [33X[0;6Ythe   non-split   extension   [23X0\rightarrow   \mathcal   O(1;\underline
        2)/F\rightarrow L\rightarrow W(1;\underline 2)^{(1)}\rightarrow 0[123X.[133X
  
  [33X[0;0YSee Theorem 5.1 of [Str].[133X
  
  
  [1X3.5-2 [33X[0;0YGeneral odd characteristic[133X[101X
  
  [33X[0;0YIf   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  [23XF[123X,  let  us  define  the  following  isomorphism  classes of
  6-dimensional non-solvable Lie algebras.[133X
  
  [31X1[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\oplus\mbox{sl}(2,F) [123X.[133X
  
  [31X2[131X   [33X[0;6Y[23X\mbox{sl}(2,F_{q^2})[123X where [23XF=F_q[123X;[133X
  
  [31X3[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\oplus  K[123X  where  [23XK[123X  is  a  solvable  Lie  algebra  with
        dimension 3;[133X
  
  [31X4[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\ltimes    (V(0)\oplus   V(1))[123X   where   [23XV(i)[123X   is   the
        [23X(i+1)[123X-dimensional irreducible [23X\mbox{sl}(2,F)[123X-module;[133X
  
  [31X5[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\ltimes V(2)[123X where [23XV(2)[123X is the [23X3[123X-dimensional irreducible
        [23X\mbox{sl}(2,F)[123X-module;[133X
  
  [31X6[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\ltimes(V(1)\oplus  C(L))\cong  \mbox{sl}(2,F)\ltimes  H[123X
        where [23XH[123X is the Heisenberg Lie algebra;[133X
  
  [31X7[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\ltimes   K[123X   where   [23XK=Fd\oplus   K^{(1)}[123X,  [23XK^{(1)}[123X  is
        2-dimensional  abelian,  isomorphic,  as  an [23X\mbox{sl}(2,F)[123X-module, to
        [23XV(1)[123X, [23X[\mbox{sl}(2,F),d]=0[123X, and, for all [23Xv\in K[123X, [23X[d,v]=v[123X;[133X
  
  [33X[0;0YIf  the  characteristic  of  [23XF[123X  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.[133X
  
  
  [1X3.5-3 [33X[0;0YCharacteristic 3[133X[101X
  
  [33X[0;0YIf  the  characteristic  of  the  field [23XF[123X is 3, then, besides the classes in
  Section [14X3.5-2[114X, we also obtain the following isomorphism classes.[133X
  
  [31X1[131X   [33X[0;6Y[23X\mbox{sl}(2,F)\ltimes   V(2,\chi)[123X   where   [23X\chi[123X  is  a  3-dimensional
        character  of  [23X\mbox{sl}(2,F)[123X.  Each  such character is described by a
        field  element  [23X\xi[123X  such  that  [23XT^3+T^2-\xi[123X  has  a  root  in  [23XF[123X; see
        Proposition 3.5 of [Str] for more details.[133X
  
  [31X2[131X   [33X[0;6Y[23XW(1;\underline  1)\ltimes\mathcal  O(1;\underline  1)[123X  where  [23X\mathcal
        O(1;\underline 1)[123X is considered as an abelian Lie algebra.[133X
  
  [31X3[131X   [33X[0;6Y[23XW(1;\underline  1)\ltimes\mathcal  O(1;\underline  1)^*[123X where [23X\mathcal
        O(1;\underline  1)^*[123X  is the dual of [23X\mathcal O(1;\underline 1)[123X and it
        is considered as an abelian Lie algebra.[133X
  
  [31X4[131X   [33X[0;6YOne  of  the  two  6-dimensional  central  extensions of the non-split
        extension        [23X0\rightarrow       V(1)\rightarrow       L\rightarrow
        \mbox{sl}(2,F)\rightarrow  0[123X;  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.[133X
  
  [31X5[131X   [33X[0;6YOne   of   the   two   non-split  extensions  [23X0\rightarrow\mbox{rad  }
        L\rightarrow   L\rightarrow   L/\mbox{rad  }  L\rightarrow  0[123X  with  a
        5-dimensional ideal; see Theorem 5.4 of [Str].[133X
  
  [33X[0;0YWe 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).[133X
  
  
  [1X3.5-4 [33X[0;0YCharacteristic 5[133X[101X
  
  [33X[0;0YIf  the  characteristic  of  the  field [23XF[123X is 5, then, besides the classes in
  Section [14X3.5-2[114X, we also obtain the following isomorphism classes.[133X
  
  [31X1[131X   [33X[0;6Y[23XW(1;\underline 1)\oplus F[123X.[133X
  
  [31X2[131X   [33X[0;6YThe non-split central extension [23X0\rightarrow F\rightarrow L\rightarrow
        W(1;\underline 1)\rightarrow 0[123X.[133X
  
  
  [1X3.6 [33X[0;0YDescription of the simple Lie algebras[133X[101X
  
  [33X[0;0YIf  [3XF[103X  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])[103X   and
  [3XNonSolvableLieAlgebra(F,[6,2])[103X  (see [3XNonSolvableLieAlgebra[103X for the details).
  Over  a field of characteristic 5, there is an additional simple Lie algebra
  with  dimension 5, namely [3XNonSolvableLieAlgebra(F,[5,3])[103X. These are the only
  isomorphism  types of simple Lie algebras over finite fields up to dimension
  6.[133X
  
  [33X[0;0YIn  addition  to  the  algebras  above  the  package contains the simple Lie
  algebras  of  dimension  between 7 and 9 over [3XGF(2)[103X. These Lie algebras were
  determined by [VL06] and can be described as follows.[133X
  
  [33X[0;0YThere  are two isomorphism classes of 7-dimensional Lie algebras over [3XGF(2)[103X.
  In a basis [22Xb1,...,b7[122X the non-trivial products in the first algebra are[133X
  
  [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;
  
  [33X[0;0Yand those in the second are[133X
  
  [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.
  
  [33X[0;0YOver  [3XGF(2)[103X  there  are  two  isomorphism  types of simple Lie algebras with
  dimension  8.  In the basis [22Xb1,...,b8[122X the non-trivial products for the first
  one are[133X
  
  [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;
  
  [33X[0;0Yand for the second one they are[133X
  
  [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.
  
  [33X[0;0YThe  non-trivial products for the unique simple Lie algebra with dimension 9
  over [3XGF(2)[103X are as follows:[133X
  
  [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 [33X[0;0YDescription of the solvable and nilpotent Lie algebras[133X[101X
  
  [33X[0;0YIn  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] and
  [CdGS11].  In  dimension  2  there  are  just  two  classes  of solvable Lie
  algebras:[133X
  
  [30X    [33X[0;6Y[22XL_2^1[122X: The Abelian Lie algebra.[133X
  
  [30X    [33X[0;6Y[22XL_2^2[122X: [22X[x_2,x_1]=x_1[122X.[133X
  
  [33X[0;0YWe have the following solvable Lie algebras of dimension 3:[133X
  
  [30X    [33X[0;6Y[22XL_3^1[122X The Abelian Lie algebra.[133X
  
  [30X    [33X[0;6Y[22XL_3^2[122X [22X[x_3,x_1]=x_1, [x_3,x_2]=x_2[122X.[133X
  
  [30X    [33X[0;6Y[22XL_3^3(a)[122X [22X[x_3,x_1]=x_2, [x_3,x_2]=ax_1+x_2[122X.[133X
  
  [30X    [33X[0;6Y[22XL_3^4(a)[122X [22X[x_3,x_1]=x_2, [x_3,x_2]=ax_1.[122X[133X
  
  [33X[0;0YAnd the following solvable Lie algebras of dimension 4:[133X
  
  [30X    [33X[0;6Y[22XL_4^1[122X The Abelian Lie algebra.[133X
  
  [30X    [33X[0;6Y[22XL_4^2[122X [22X[x_4,x_1]=x_1, [x_4,x_2]=x_2, [x_4,x_3]=x_3.[122X[133X
  
  [30X    [33X[0;6Y[22XL_4^3(a)[122X [22X[x_4,x_1]=x_1, [x_4,x_2]=x_3, [x_4,x_3]=-ax_2 +(a+1)x_3[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^4[122X [22X[x_4,x_2]=x_3, [x_4,x_3]= x_3[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^5[122X [22X[x_4,x_2]=x_3[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^6(a,b)[122X [22X[x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2+x_3[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^7(a,b)[122X [22X[x_4,x_1] = x_2, [x_4,x_2]=x_3, [x_4,x_3] = ax_1+bx_2.[122X[133X
  
  [30X    [33X[0;6Y[22XL_4^8[122X [22X[x_1,x_2]=x_2, [x_3,x_4]=x_4[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^9(a)[122X   [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^10(a)[122X    [22X[x_4,x_1]    =    x_2,   [x_4,x_2]=ax_1,   [x_3,x_1]=x_1,
        [x_3,x_2]=x_2[122X Condition on F: the characteristic of F is 2.[133X
  
  [30X    [33X[0;6Y[22XL_4^11(a,b)[122X  [22X[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[122X. Condition on F: the characteristic of F
        is 2.[133X
  
  [30X    [33X[0;6Y[22XL_4^12[122X   [22X[x_4,x_1]   =   x_1,   [x_4,x_2]=2x_2,   [x_4,x_3]   =   x_3,
        [x_3,x_1]=x_2[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^13(a)[122X  [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XL_4^14(a)[122X [22X[x_4,x_1] = ax_3, [x_4,x_3]=x_1, [x_3,x_1]=x_2[122X.[133X
  
  [33X[0;0YNilpotent of dimension 5:[133X
  
  [30X    [33X[0;6Y[22XN_5,1[122X Abelian.[133X
  
  [30X    [33X[0;6Y[22XN_5,2[122X [22X[x_1,x_2]=x_3[122X.[133X
  
  [30X    [33X[0;6Y[22XN_5,3[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4[122X.[133X
  
  [30X    [33X[0;6Y[22XN_5,4[122X [22X[x_1,x_2]=x_5, [x_3,x_4]=x_5[122X.[133X
  
  [30X    [33X[0;6Y[22XN_5,5[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]=x_5[122X.[133X
  
  [30X    [33X[0;6Y[22XN_5,6[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_5[122X.[133X
  
  [30X    [33X[0;6Y[22XN_5,7[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5[122X.[133X
  
  [30X    [33X[0;6Y[22XN_5,8[122X [22X[x_1,x_2]=x_4, [x_1,x_3]=x_5[122X.[133X
  
  [30X    [33X[0;6Y[22XN_5,9[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_2,x_3]=x_5[122X.[133X
  
  [33X[0;0YWe get nine 6-dimensional nilpotent Lie algebras denoted [22XN_6,k[122X for [22Xk=1,...,9[122X
  that  are  the  direct  sum  of  [22XN_5,k[122X  and  a  1-dimensional abelian ideal.
  Subsequently we get the following Lie algebras.[133X
  
  [30X    [33X[0;6Y[22XN_6,10[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_6, [x_4,x_5]=x_6.[122X[133X
  
  [30X    [33X[0;6Y[22XN_6,11[122X  [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,12[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_5]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,13[122X  [22X[x_1,x_2]=x_3,  [x_1,x_3]=x_5,  [x_1,x_5]=x_6,  [x_2,x_4]=x_5,
        [x_3,x_4]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,14[122X  [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,15[122X  [22X[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_5, [x_2,x_4]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,16[122X  [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,17[122X  [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,18[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_1,x_5]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,19(a)[122X  [22X[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_6,
        [x_3,x_5]=a x_6[122X, for [22Xa≠0[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,20[122X [22X[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,21(a)[122X  [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_6, [x_2,x_3]=x_5,
        [x_2,x_5]= a x_6[122X, for [22Xa≠0[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,22(a)[122X    [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,23[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6, [x_2,x_4]= x_5[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,24(a)[122X     [22X[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[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,25[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_4]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,26[122X [22X[x_1,x_2]=x_4, [x_1,x_3]=x_5, [x_2,x_3]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,27[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_2,x_4]= x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,28[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_4, [x_1,x_4]=x_5, [x_2,x_3]=x_6[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,29[122X [22X[x_1,x_2]=x_3, [x_1,x_3]=x_5, [x_1,x_5]=x_6, [x_2,x_4]=x_5+x_6,
        [x_3,x_4]=x_6[122X, only over fields of characteristic [22X2[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,30[122X  [22X[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_5+x_6,  [x_2,x_4]=x_6[122X,  only over fields of characteristic
        [22X2[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,31(a)[122X     [22X[x_1,x_2]=x_3,     [x_1,x_3]=x_4,     [x_1,x_4]=    x_5,
        [x_2,x_3]=x_5+a  x_6,  [x_2,x_5]=x_6,  [x_3,x_4]=x_6[122X, for [22Xa≠0[122X and only
        over fields of characteristic [22X2[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,32(a)[122X  [22X[x_1,x_2]=x_3,  [x_1,x_3]=x_4,  [x_1,x_4]= x_5, [x_2,x_3]=a
        x_6,  [x_2,x_5]=x_6,  [x_3,x_4]=x_6[122X,  for  [22Xa≠0[122X and only over fields of
        characteristic [22X2[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,33[122X  [22X[x_1,x_2]=x_4,  [x_1,x_3]=x_5,  [x_2,x_5]=x_6,  [x_3,x_4]=x_6[122X,
        only over fields of characteristic [22X2[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,34[122X  [22X[x_1,x_2]=x_3,  [x_1,x_3]=x_4,  [x_1,x_5]=x_6,  [x_2,x_3]=x_5,
        [x_2,x_4]=x_6[122X, only over fields of characteristic [22X2[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,35(a)[122X    [22X[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+x_6[122X, only over fields of characteristic [22X2[122X.[133X
  
  [30X    [33X[0;6Y[22XN_6,36(a)[122X    [22X[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+x_6[122X,  only over fields of characteristic
        [22X2[122X.[133X
  
  [33X[0;0YIn  [CdGS11], the Lie algebras [22XN_5,k[122X are denoted by [22XL_5,k[122X for all [22Xk=1,...,9[122X.
  Similarly, the Lie algebras [22XN_6,k[122X or [22XN_6,k(a)[122X, where [22Xk=1,...,36[122X, are denoted
  by [22XL_6,k[122X or [22XL_6,k(a)[122X if [22Xk=1,...,28[122X and by [22XL_6,k-28^(2)[122X or [22XL_6,k-28^(2)(a)[122X if
  [22Xk=29,...,36[122X.[133X
  
