  
  [1X4 [33X[0;0YFinite Order Automorphisms and [22Xθ[122X[101X[1X-Groups[133X[101X
  
  [33X[0;0YThis  chapter  contains functions for creating and working with finite order
  automorphisms of simple Lie algebras (or, more precisely, representatives of
  the conjugacy classes of such automorphisms).[133X
  
  [33X[0;0YNB:  such automorphisms are not created for a given Lie algebra, but the Lie
  algebra  is  constructed  at the same time as the automorphism. This because
  the base field may need extending (it needs enough roots of unity).[133X
  
  [33X[0;0YAs  noted above the functions give representatives of the conjugacy classes,
  in  the  automorphism  group  of the underlying Lie algebra, of finite order
  automorphisms.  Such  conjugacy  classes  are  classified  in  terms  of Kac
  diagrams.  Roughly,  this  works  as  follows. A finite order automorphism [22Xf[122X
  corresponds   to   a  diagram  automorphism  of  order  [22Xd=1,2,3[122X.  The  inner
  automorphisms  correspond  to  a  diagram automorphism of order 1, the outer
  automorphisms to a diagram automorphism of order 2 or 3. Let [22XL_0, L_1[122X denote
  the eigenspaces of the underlying Lie algebra [22XL[122X, with respect to the diagram
  automorphism, respectively corresponding to the eigenvalues 1 and [22Xw[122X (where [22Xw[122X
  is  a primitive [22Xd[122X-th root of unity). (In case of [22Xd=1[122X, we have [22XL_0=L[122X, [22XL_1=0[122X.)
  Then  [22XL_0[122X  is semisimple and we choose a set of canonical generators of [22XL_0[122X,
  denoted  [22Xx_i[122X,  [22Xy_i[122X,  [22Xh_i[122X, for [22Xi=1,...,s[122X. Moreover, [22XL_1[122X is an [22XL_0[122X-module. Let
  [22Xx_0[122X  be the lowest weight vector in [22XL_1[122X. (If [22Xd=1[122X then [22Xx_0[122X will be the lowest
  (negative) root vector.) Let [22Xα_i[122X for [22Xi=0,...,s[122X be the roots corresponding to
  [22Xx_i[122X,  with respect to the subalgebra spanned by the [22Xh_i[122X. Let [22XC[122X be the Cartan
  matrix  of  these  roots.  The  rows of [22XC[122X are linearly dependent. The Dynkin
  diagram  of  [22XC[122X  is labeled with integers [22Xa_i[122X with greatest common divisor 1,
  that  form  the  coefficients  of  a  linear  dependency  of  the rows of [22XC[122X.
  Furthermore,  the  [22Xx_i[122X  generate  [22XL[122X  and  the automorphism [22Xf[122X is described by
  [22Xf(x_i) = v^s_i x_i[122X, where the non-negative integers [22Xs_i[122X have greatest common
  divisor  1, and are such that [22Xm=d∑ a_i s_i[122X is the order of [22Xf[122X, and where [22Xv[122X is
  a  primitive  [22Xm[122X-th  order  root  of  unity.  Now  the  Kac  diagram  of  the
  automorphism [22Xf[122X is the Dynkin diagram of [22XC[122X, labelled with the labels [22Xs_i[122X.[133X
  
  
  [1X4.1 [33X[0;0YThe functions[133X[101X
  
  [1X4.1-1 FiniteOrderInnerAutomorphisms[101X
  
  [33X[1;0Y[29X[2XFiniteOrderInnerAutomorphisms[102X( [3Xtype[103X, [3Xrank[103X, [3Xm[103X ) [32X operation[133X
  
  [33X[0;0YLet  [22XL[122X  be  the  simple Lie algebra of type [3Xtype[103X and rank [3Xrank[103X. The function
  returns representatives of the conjugacy classes of inner automorphisms of [22XL[122X
  of order [3Xm[103X. As noted also in the introduction to this chapter, this function
  constructs the Lie algebra as well as the automorphisms (and the Lie algebra
  is  accessible  through  the  source of these automorphisms). The reason for
  this  is  that  depending  on the order of the automorphisms, the base field
  needs certain roots of unity.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= FiniteOrderInnerAutomorphisms("E",6,3);[127X[104X
    [4X[28X[ [ v.72, v.1, v.2, v.3, v.4, v.5, v.6 ] -> [ (E(3))*v.72, (E(3)^2)*v.1, v.2, [128X[104X
    [4X[28X      v.3, v.4, v.5, v.6 ], [ v.72, v.1, v.2, v.3, v.4, v.5, v.6 ] -> [128X[104X
    [4X[28X    [ v.72, (E(3))*v.1, (E(3))*v.2, v.3, v.4, v.5, v.6 ], [128X[104X
    [4X[28X  [ v.72, v.1, v.2, v.3, v.4, v.5, v.6 ] -> [ (E(3))*v.72, v.1, (E(3))*v.2, [128X[104X
    [4X[28X      v.3, v.4, v.5, v.6 ], [ v.72, v.1, v.2, v.3, v.4, v.5, v.6 ] -> [128X[104X
    [4X[28X    [ v.72, v.1, v.2, v.3, (E(3))*v.4, v.5, v.6 ], [128X[104X
    [4X[28X  [ v.72, v.1, v.2, v.3, v.4, v.5, v.6 ] -> [ (E(3))*v.72, (E(3))*v.1, v.2, [128X[104X
    [4X[28X      v.3, v.4, v.5, (E(3))*v.6 ] ][128X[104X
    [4X[25Xgap>[125X [27XSource(f[1]);[127X[104X
    [4X[28X<Lie algebra of dimension 78 over CF(3)>[128X[104X
  [4X[32X[104X
  
  [1X4.1-2 FiniteOrderOuterAutomorphisms[101X
  
  [33X[1;0Y[29X[2XFiniteOrderOuterAutomorphisms[102X( [3Xtype[103X, [3Xrank[103X, [3Xm[103X, [3Xd[103X ) [32X operation[133X
  
  [33X[0;0YLet  [22XL[122X  be  the  simple Lie algebra of type [3Xtype[103X and rank [3Xrank[103X. The function
  returns representatives of the conjugacy classes of outer automorphisms of [22XL[122X
  of order [3Xm[103X, corresponding to a diagram automorphism of order [3Xd[103X.[133X
  
  [1X4.1-3 Order[101X
  
  [33X[1;0Y[29X[2XOrder[102X( [3Xf[103X ) [32X attribute[133X
  
  [33X[0;0YHere [3Xf[103X is a finite order automorphism. This returns its order.[133X
  
  [1X4.1-4 KacDiagram[101X
  
  [33X[1;0Y[29X[2XKacDiagram[102X( [3Xf[103X ) [32X attribute[133X
  
  [33X[0;0YHere [3Xf[103X is a finite order automorphism. This returns its Kac diagram. This is
  a record with three components: [3XCM[103X, which is the Cartan matrix of the Dynkin
  diagram,  [3Xlabels[103X  the integers with gcd equal to 1 that are the coefficients
  of  a linear dependency of the rows of [3XCM[103X, and [3Xweights[103X that are the integers
  [22Xs_i[122X that define the automorphism.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= FiniteOrderOuterAutomorphisms( "A", 5, 4, 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xr:= KacDiagram( f[1] );[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X  CM := [ [ 2, 0, -1, 0 ], [ 0, 2, -1, 0 ], [ -1, -1, 2, -1 ], [128X[104X
    [4X[28X      [ 0, 0, -2, 2 ] ], labels := [ 1, 1, 2, 1 ], weights := [ 1, 1, 0, 0 ] )[128X[104X
    [4X[25Xgap>[125X [27Xr.labels*r.CM;      [127X[104X
    [4X[28X[ 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [1X4.1-5 Grading[101X
  
  [33X[1;0Y[29X[2XGrading[102X( [3Xf[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3Xf[103X  is  a  finite order automorphism of order [22Xm[122X. This returns a list of
  length  [22Xm[122X.  The  [22Xi[122X-th  element  contains a basis of the eigenspace of [3Xf[103X with
  eigenvalue [22Xv^i[122X, where [22Xv[122X is a primitive [22Xm[122X-th root of unity (i.e., [3Xv=E(m)[103X).[133X
  
  [1X4.1-6 NilpotentOrbitsOfThetaRepresentation[101X
  
  [33X[1;0Y[29X[2XNilpotentOrbitsOfThetaRepresentation[102X( [3Xf[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XNilpotentOrbitsOfThetaRepresentation[102X( [3XL[103X, [3Xd[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3Xf[103X  is  an  automorphism  of  a simple Lie algebra [22XL[122X of order [22Xm[122X. Then [3Xf[103X
  defines  a  grading  on [22XL[122X. Let the homogeneous components of this grading be
  denoted  [22XL_i[122X  for  [22Xi=0,...,m-1[122X.  Let  [22XG_0[122X  be the group corresponding to [22XL_0[122X
  (i.e.,  the  connected  subgroup  of the adjoint group of [22XL[122X with Lie algebra
  [22XL_0[122X). This function computes representatives for the nilpotent orbits of [22XG_0[122X
  acting  on  [22XL_1[122X.  The  output  is  a  list  of  triples.  Each  triple is an
  [22Xsl_2[122X-triple  [22X(y,h,x)[122X, with [22Xh∈ L_0[122X, [22Xx∈ L_1[122X (the representative of the orbit),
  and  [22Xy∈  L_m-1[122X.  The  element  [22Xh[122X also lies in the dominant Weyl chamber of a
  Cartan subalgebra of [22XL_0[122X. Finally we note that all elements lie in [3XSource( f
  )[103X.[133X
  
  [33X[0;0YIt  is  possible  to  add an extra optional argument: [3Xmethod:= "Carrier"[103X, or
  [3Xmethod:=  "WeylOrbit"[103X.  Then  a  method  based  on  finding carrier algebras
  (respectively,  computing  orbits  under  the  Weyl  group) is chosen. If no
  optional  argument  is chosen, then the system will make its own choice. (In
  the  case of outer automorphisms, currently the only available method is the
  one based on orbits of the Weyl group.) The method based on carrier algebras
  tends to work better for the higher order automorphisms.[133X
  
  [33X[0;0YThis  function prints some information on what it is doing to the info class
  [3XInfoSLA[103X.  In  order  to  suppress  these  messages  one can do [3XSetInfoLevel(
  InfoSLA, 1 );[103X.[133X
  
  [33X[0;0YIn the two-argument version, the first argument [3XL[103X has to be a semisimple Lie
  algebra,  and  the second argument [3Xd[103X a list of non-negative integers. Then [3XL[103X
  is  [22XZ[122X-graded  by giving the root space corresponding to the [22Xi[122X-th simple root
  the degree [3Xd[i][103X. Apart from this the function works the same in this case as
  in the one-argument version.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X# reset random state to ensure the output of this example match[127X[104X
    [4X[25Xgap>[125X [27XReset(GlobalMersenneTwister, 1);;[127X[104X
    [4X[25Xgap>[125X [27Xf:= FiniteOrderInnerAutomorphisms( "D", 5, 3 );;   [127X[104X
    [4X[25Xgap>[125X [27Xs:= NilpotentOrbitsOfThetaRepresentation( f[2] : method:= "Carrier" );;[127X[104X
    [4X[28X#I  Selected carrier algebra method.[128X[104X
    [4X[28X#I  Constructed 123 root bases of possible flat subalgebras, now checking them...[128X[104X
    [4X[28X#I  Obtained 30 Cartan elements, weeding out equivalent copies...[128X[104X
    [4X[25Xgap>[125X [27XLength(s);[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27Xs[4];[127X[104X
    [4X[28X[ v.14+v.15+v.38, (-2)*v.41+(-1)*v.42, v.18+v.34+v.35 ][128X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("E",6,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XNilpotentOrbitsOfThetaRepresentation( L, [0,1,0,0,0,0] );[127X[104X
    [4X[28X#I  Selected Weyl orbit method.[128X[104X
    [4X[28X#I  Constructed a Weyl transversal of 72 elements.[128X[104X
    [4X[28X#I  Obtained 5 Cartan elements, weeding out equivalent copies...[128X[104X
    [4X[28X[ [ v.65+v.66+v.67, (2)*v.73+(3)*v.74+(4)*v.75+(6)*v.76+(4)*v.77+(2)*v.78, [128X[104X
    [4X[28X      v.29+v.30+v.31 ], [128X[104X
    [4X[28X  [ (2)*v.55+(2)*v.66, (2)*v.73+(4)*v.74+(4)*v.75+(6)*v.76+(4)*v.77+(2)*v.78, [128X[104X
    [4X[28X      v.19+v.30 ],[128X[104X
    [4X[28X  [ v.66+v.70, (2)*v.73+(2)*v.74+(3)*v.75+(4)*v.76+(3)*v.77+(2)*v.78, [128X[104X
    [4X[28X      v.30+v.34 ], [ v.71, v.73+v.74+(2)*v.75+(3)*v.76+(2)*v.77+v.78, v.35 ] ][128X[104X
  [4X[32X[104X
  
  [1X4.1-7 ClosureDiagram[101X
  
  [33X[1;0Y[29X[2XClosureDiagram[102X( [3XL[103X, [3Xf[103X, [3Xs[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XClosureDiagram[102X( [3XL[103X, [3Xd[103X, [3Xs[103X ) [32X operation[133X
  
  [33X[0;0YHere [3Xf[103X is an automorphism of a simple Lie algebra [22XL[122X of order [22Xm[122X, and [3Xs[103X a list
  of  [22Xsl_2[122X-triples  [22X(y,h,x)[122X,  with [22Xh∈ L_0[122X, [22Xx∈ L_1[122X (for instance as computed by
  the previous function), corresponding to nilpotent orbits in [22XL_1[122X.[133X
  
  [33X[0;0YThis  function  computes  the Hasse diagram of the closures of the nilpotent
  orbits. The output is a record with two components: [3Xdiag[103X (which is a list of
  2-tuples;  a  tuple  [3X[  i, j ][103X means that orbit number [3Xi[103X is contained in the
  closure  of  orbit  number  [3Xj[103X),  and [3Xsl2[103X (the same list of [22Xsl_2[122X-triples, but
  sorted  according  to  decreasing  dimension,  i.e., the highest dimensional
  orbit  comes first). The numbering used in the tuples in [3Xdiag[103X corresponds to
  the order in which the orbits appear in the component [3Xsl2[103X.[133X
  
  [33X[0;0YDuring  the  execution  of  the  program  a message is printed. This message
  either  states  that  all  inclusions have been proved, or lists a number of
  possible  inclusions,  for  which  it  could  not  be  proved  with absolute
  certainty  that  these do not occur. This is due to the randomised nature of
  the  algorithm:  if the algorithm finds an inclusion, then this inclusion is
  certain. However, sometimes a non-inclusion can only be estabished by random
  methods,  which means that it is possible that there is an inclusion without
  the  program  finding  it.  (This however, is very unlikely, and in practice
  almost  never  happens.)  Now  showing  that  a  non-inclusion  really  is a
  non-inclusion  can  be  done by computing the ranks of certain matrices with
  polynomial  entries.  In  principle  [5XGAP[105X  can  do  this; however, the system
  certainly  is  not  very  strong  at  it.  Therefore, as optional argument a
  filename can be given, by [3Xfilenm:= "file.m"[103X. If this argument is present the
  program prints a Magma script in the file, which can be loaded directly into
  the  computer  algebra  system Magma. If the output is always true, then all
  non-inclusions  are  proved.  If  there are non non-inclusions to be proved,
  then the file is not written.[133X
  
  [33X[0;0YIn  the  second  version,  the  second  argument [3Xd[103X is a list of non-negative
  integers.  Then  [3XL[103X is [22XZ[122X-graded by giving the root space corresponding to the
  [22Xi[122X-th  simple root the degree [3Xd[i][103X. Apart from this the function works in the
  same way.[133X
  
  [33X[0;0YWe  note  that the adjoint representation can be obtained by giving a [3Xd[103X that
  eintirely consists of zeros.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= FiniteOrderInnerAutomorphisms( "E", 8, 8 );;  [127X[104X
    [4X[25Xgap>[125X [27Xh:= f[8];;[127X[104X
    [4X[25Xgap>[125X [27Xsl2:= NilpotentOrbitsOfThetaRepresentation(h);;  [127X[104X
    [4X[28X#I  Selected carrier algebra method.[128X[104X
    [4X[28X#I  Constructed 2782 root bases of possible flat subalgebras, now checking them...[128X[104X
    [4X[28X#I  Obtained 58 Cartan elements, weeding out equivalent copies...[128X[104X
    [4X[25Xgap>[125X [27XLength(sl2);[127X[104X
    [4X[28X27[128X[104X
    [4X[25Xgap>[125X [27XL:= Source(h);;                    [127X[104X
    [4X[25Xgap>[125X [27Xr:= ClosureDiagram( L, h, sl2 );;  [127X[104X
    [4X[28X#I  All (non-) inclusions proved![128X[104X
    [4X[25Xgap>[125X [27Xr.diag;[127X[104X
    [4X[28X[ [ 2, 1 ], [ 3, 1 ], [ 4, 2 ], [ 4, 3 ], [ 5, 1 ], [ 6, 5 ], [ 7, 2 ], [128X[104X
    [4X[28X  [ 7, 5 ], [ 8, 4 ], [ 9, 4 ], [ 9, 7 ], [ 10, 6 ], [ 10, 7 ], [ 11, 3 ], [128X[104X
    [4X[28X  [ 11, 6 ], [ 12, 7 ], [ 13, 9 ], [ 13, 10 ], [ 13, 11 ], [ 14, 9 ], [128X[104X
    [4X[28X  [ 14, 12 ], [ 15, 8 ], [ 15, 9 ], [ 16, 6 ], [ 17, 10 ], [ 17, 12 ], [128X[104X
    [4X[28X  [ 17, 16 ], [ 18, 13 ], [ 18, 16 ], [ 19, 13 ], [ 19, 15 ], [ 20, 11 ], [128X[104X
    [4X[28X  [ 20, 16 ], [ 21, 14 ], [ 21, 17 ], [ 21, 18 ], [ 22, 14 ], [ 22, 15 ], [128X[104X
    [4X[28X  [ 23, 18 ], [ 23, 20 ], [ 24, 18 ], [ 24, 19 ], [ 25, 21 ], [ 25, 22 ], [128X[104X
    [4X[28X  [ 25, 24 ], [ 26, 23 ], [ 26, 24 ], [ 27, 21 ], [ 27, 23 ] ][128X[104X
    [4X[25Xgap>[125X [27X# Now we do the adjoint representation of the Lie algebra of type F4:[127X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("F",4,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27Xo:= NilpotentOrbits(L);;[127X[104X
    [4X[25Xgap>[125X [27Xsl2:= List( o, SL2Triple );;[127X[104X
    [4X[25Xgap>[125X [27Xr:= ClosureDiagram( L, [0,0,0,0], sl2 );;      [127X[104X
    [4X[28X#I  All (non-) inclusions proved![128X[104X
    [4X[25Xgap>[125X [27Xr.diag;[127X[104X
    [4X[28X[ [ 2, 1 ], [ 3, 2 ], [ 4, 3 ], [ 5, 3 ], [ 6, 4 ], [ 6, 5 ], [ 7, 6 ], [128X[104X
    [4X[28X  [ 8, 7 ], [ 9, 7 ], [ 10, 8 ], [ 10, 9 ], [ 11, 8 ], [ 12, 10 ], [128X[104X
    [4X[28X  [ 13, 11 ], [ 13, 12 ], [ 14, 13 ], [ 15, 14 ] ][128X[104X
  [4X[32X[104X
  
  [1X4.1-8 CarrierAlgebra[101X
  
  [33X[1;0Y[29X[2XCarrierAlgebra[102X( [3XL[103X, [3Xf[103X, [3Xe[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XCarrierAlgebra[102X( [3XL[103X, [3Xd[103X, [3Xe[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3Xf[103X  is  an  automorphism  of a simple Lie algebra [22XL[122X of order [22Xm[122X, and [3Xe[103X a
  nilpotent  element  of  [22XL_1[122X. This function returns the carrier algebra of [3Xe[103X.
  This  is  a  [22XZ[122X-graded semisimple subalgebra [22XK[122X of [22XL[122X, such that [3Xe[103X lies in [22XK_1[122X.
  For the precise definition we refer to [Vin79], [Vin75]. The output is given
  in  the  form  of  a record, with three components: [3Xg0[103X, a basis of [22XK_0[122X, [3Xgp[103X a
  list  containing bases of [22XK_1[122X, [22XK_2[122X and so on, and [3Xgn[103X a list containing bases
  of [22XK_-1[122X, [22XK_-2[122X and so on.[133X
  
  [33X[0;0YIn  the  second  version,  the  second  argument [3Xd[103X is a list of non-negative
  integers.  Then  [3XL[103X is [22XZ[122X-graded by giving the root space corresponding to the
  [22Xi[122X-th  simple root the degree [3Xd[i][103X. Apart from this the function works in the
  same way.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= FiniteOrderInnerAutomorphisms( "F", 4, 5 );;[127X[104X
    [4X[25Xgap>[125X [27Xh:= f[4];;[127X[104X
    [4X[25Xgap>[125X [27Xsl2:= NilpotentOrbitsOfThetaRepresentation( h );;  [127X[104X
    [4X[28X#I  Selected Weyl orbit method.[128X[104X
    [4X[28X#I  Constructed a Weyl transversal of 144 elements.[128X[104X
    [4X[28X#I  Constructed 621 Cartan elements to be checked.[128X[104X
    [4X[25Xgap>[125X [27XL:= Source(h);   [127X[104X
    [4X[28X<Lie algebra of dimension 52 over CF(5)>[128X[104X
    [4X[25Xgap>[125X [27Xr:=CarrierAlgebra( L, h, sl2[1][3] );   [127X[104X
    [4X[28Xrec( g0 := [ v.49+(2)*v.50+(2)*v.51+(3)*v.52, v.50+(1/2)*v.51+v.52 ], [128X[104X
    [4X[28X  gn := [ [ v.24, v.33 ], [ v.21 ], [ v.15 ] ], [128X[104X
    [4X[28X  gp := [ [ v.9, v.48 ], [ v.45 ], [ v.39 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XK:= Subalgebra( L, Concatenation( r.g0, Flat(r.gp), Flat(r.gn) ) );[127X[104X
    [4X[28X<Lie algebra over CF(5), with 10 generators>[128X[104X
    [4X[25Xgap>[125X [27XSemiSimpleType( K );[127X[104X
    [4X[28X"B2"[128X[104X
  [4X[32X[104X
  
  [1X4.1-9 CartanSubspace[101X
  
  [33X[1;0Y[29X[2XCartanSubspace[102X( [3Xf[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3Xf[103X  is  an  automorphism  of  a simple Lie algebra [22XL[122X of order [22Xm[122X. Then [3Xf[103X
  defines  a  grading  on [22XL[122X. Let the homogeneous components of this grading be
  denoted  [22XL_i[122X  for  [22Xi=0,...,m-1[122X.  Let  [22XG_0[122X  be the group corresponding to [22XL_0[122X
  (i.e.,  the  connected  subgroup  of the adjoint group of [22XL[122X with Lie algebra
  [22XL_0[122X).  This  function  computes  a  maximal  subspace  of  [22XL_1[122X consisting of
  commuting   semisimple  elements.  (Such  a  subspace  is  called  a  [13XCartan
  subspace[113X.)[133X
  
  [33X[0;0YEvery  semisimple  orbit of [22XG_0[122X in [22XL_1[122X contains an element of a fixed Cartan
  subspace.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= FiniteOrderInnerAutomorphisms( "A", 3, 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xc:= CartanSubspace( f[3] ); [127X[104X
    [4X[28X<vector space of dimension 1 over CF(3)>[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( Basis( c ) );[127X[104X
    [4X[28X[ v.1+v.5+v.12 ][128X[104X
  [4X[32X[104X
  
