H* Octonions Multiplication Table *L
QQ@1,
QQ@1,
QQ@1,
QQ@1,
QQ@1,
QQ@1,
QQ@1,
QQ@1,
QQ@2,
QQ@2,
QQ@2,
QQ@2,
QQ@2,
QQ@2,
QQ@2,
QQ@2,
QQ@3,
QQ@3,
QQ@3,
QQ@3,
QQ@3,
QQ@3,
QQ@3,
QQ@3,
QQ@4,
QQ@4,
QQ@4,
QQ@4,
QQ@4,
QQ@4,
QQ@4,
QQ@4,
QQ@5,
QQ@5,
QQ@5,
QQ@5,
QQ@5,
QQ@5,
QQ@5,
QQ@5,
QQ@6,
QQ@6,
QQ@6,
QQ@6,
QQ@6,
QQ@6,
QQ@6,
QQ@6,
QQ@7,
QQ@7,
QQ@7,
QQ@7,
QQ@7,
1D
2D
3D
4D
5D
6D
7D
8D
1D
2D
3D
4D
5D
6D
7D
8D
1D
2D
3D
4D
5D
6D
7D
8D
1D
2D
3D
4D
5D
6D
7D
8D
1D
2D
3D
4D
5D
6D
7D
8D
1D
2D
3D
4D
5D
6D
7D
8D
1D
2D
3D
4D
5D
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
e@1D;
e@2D;
e@3D;
e@4D;
e@5D;
e@6D;
e@7D;
e@8D;
e@2D;
- e@1D;
e@5D;
e@8D;
- e@3D;
e@7D;
- e@6D;
- e@4D;
e@3D;
- e@5D;
- e@1D;
e@6D;
e@2D;
- e@4D;
e@8D;
- e@7D;
e@4D;
- e@8D;
- e@6D;
- e@1D;
e@7D;
e@3D;
- e@5D;
e@2D;
e@5D;
e@3D;
- e@2D;
- e@7D;
- e@1D;
e@8D;
e@4D;
- e@6D;
e@6D;
- e@7D;
e@4D;
- e@3D;
- e@8D;
- e@1D;
e@2D;
e@5D;
e@7D;
e@6D;
- e@8D;
e@5D;
- e@4D;
2
e8.nb
QQ@7,
QQ@7,
QQ@7,
QQ@8,
QQ@8,
QQ@8,
QQ@8,
QQ@8,
QQ@8,
QQ@8,
QQ@8,
6D
7D
8D
1D
2D
3D
4D
5D
6D
7D
8D
=
=
=
=
=
=
=
=
=
=
=
- e@2D;
- e@1D;
e@3D;
e@8D;
e@4D;
e@7D;
- e@2D;
e@6D;
- e@5D;
- e@3D;
- e@1D;
H* Split Octonions for l=-1 *L
Do@QQ1@i, jD = QQ@i, jD;
QQ2@i, jD = QQ@i, jD, 8i, 8<, 8j, 8<D;
l1 = 1;
L = 82, 5, 7, 8<;
Do@QQ1@L@@iDD, L@@jDDD = l1 QQ@L@@iDD, L@@jDDD, 8i, 4<, 8j, 4<D;
l2 = 1;
L = 82, 5, 7, 8<;
Do@QQ2@L@@iDD, L@@jDDD = l2 QQ@L@@iDD, L@@jDDD, 8i, 4<, 8j, 4<D;
H* Octonions product *L
Qm@x_, y_D := Sum@Sum@x@@iDD y@@jDD QQ1@i, jD, 8i, 8<D, 8j, 8<D
QP@x_, y_D := 8Coefficient@Qm@x, yD, e@1DD, Coefficient@Qm@x, yD, e@2DD,
Coefficient@Qm@x, yD, e@3DD, Coefficient@Qm@x, yD, e@4DD,
Coefficient@Qm@x, yD, e@5DD, Coefficient@Qm@x, yD, e@6DD,
Coefficient@Qm@x, yD, e@7DD, Coefficient@Qm@x, yD, e@8DD<
Conj@x_D := 8x@@1DD, - x@@2DD, - x@@3DD, - x@@4DD, - x@@5DD, - x@@6DD, - x@@7DD, - x@@8DD<;
H* Construction of G2 *L
ee@1D
ee@2D
ee@3D
ee@4D
ee@5D
ee@6D
ee@7D
ee@8D
=
=
=
=
=
=
=
=
81,
80,
80,
80,
80,
80,
80,
80,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0,
0,
0,
0,
0,
0,
0,
1,
0,
0<;
0<;
0<;
0<;
0<;
0<;
0<;
1<;
Comm@a_, b_D := QP@a, bD - QP@b, aD;
OM@a_Integer, b_Integer, d_D :=
Comm@Comm@ee@aD, ee@bDD, dD - 3 QP@QP@ee@aD, ee@bDD, dD + 3 QP@ee@aD, QP@ee@bD, dDD;
OO@a_Integer, b_Integer, c_Integer, d_IntegerD := HComm@Comm@ee@aD, ee@bDD, ee@dDD 3 QP@QP@ee@aD, ee@bDD, ee@dDD + 3 QP@ee@aD, QP@ee@bD, [email protected]@cD;
Do@
OD = ConstantArray@0, 87, 7, 7, 7<D,
8i, 7<, 8j, 7<D;
Do@
OD@@i, j, l, kDD = OO@i + 1, j + 1, l + 1, k + 1D, 8i, 7<, 8j, 7<, 8l, 7<, 8k, 7<D;
e8.nb
Zero7 = ConstantArray@0, 87, 7<D;
Do@
If@OD@@i, jDD + OD@@j, iDD ¹ Zero7, Print@i, jDD, 8i, 7<, 8j, i<D
cc@1D = OD@@3, 2DD;
cc@2D = OD@@5, 2DD;
cc@3D = OD@@6, 2DD;
cc@4D = OD@@7, 2DD;
cc@5D = OD@@4, 3DD;
cc@6D = OD@@5, 3DD;
cc@7D = OD@@6, 3DD;
cc@8D = OD@@7, 3DD;
cc@9D = OD@@5, 4DD;
cc@10D = OD@@6, 4DD;
cc@11D = OD@@7, 4DD;
cc@12D = OD@@6, 5DD;
cc@13D = OD@@7, 5DD;
cc@14D = OD@@7, 6DD;
H* Gram-Schmidt orthonormalization of the G2 generators *L
Mg2@1D = cc@1D;
Do@Do@
AA@j, iD = 0, 8i, 14<D, 8j, 14<D
Do@Do@
AA@j, iD = - Tr@[email protected]@iDD  Tr@[email protected]@iDD, 8i, j - 1<D;
Mg2@jD = cc@jD + Sum@AA@j, iD Mg2@iD, 8i, j - 1<D, 8j, 2, 14<D
g2 = ConstantArray@0, 814, 7, 7<D;
Do@g2@@iDD = Mg2@iD  Sqrt@Abs@Tr@[email protected]@iDDDD, 8i, 14<D;
H* Structure constants of G2 *L
Co@a_, b_D := a.b - b.a;
Cg2 = ConstantArray@0, 814, 14, 14<D;
Do@Cg2@@i, j, kDD = Tr@Co@g2@@iDD, g2@@jDDD.g2@@kDDD  Tr@g2@@kDD.g2@@kDDD,
8i, 14<, 8j, 14<, 8k, 14<D;
Clear@A, B, AA, BB, U1, U2, Ad, ccomD
H* Construction of F4 *L
MT = 8 88mt@1D + mt@18D  Sqrt@3D + mt@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<,
8mt@2D, mt@3D, mt@4D, mt@5D, mt@6D, mt@7D, mt@8D, mt@9D<,
8mt@10D, mt@11D, mt@12D, mt@13D, mt@14D, mt@15D, mt@16D, mt@17D<<,
88mt@2D, - mt@3D, - mt@4D, - mt@5D, - mt@6D, - mt@7D, - mt@8D, - mt@9D<,
8- mt@1D + mt@18D  Sqrt@3D + mt@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<,
8mt@19D, mt@20D, mt@21D, mt@22D, mt@23D, mt@24D, mt@25D, mt@26D<<,
88mt@10D, - mt@11D, - mt@12D, - mt@13D, - mt@14D, - mt@15D, - mt@16D, - mt@17D<,
8mt@19D, - mt@20D, - mt@21D, - mt@22D, - mt@23D, - mt@24D, - mt@25D, - mt@26D<,
8- 2 * mt@18D  Sqrt@3D + mt@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<<<;
B@1D = 8 881, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 8- 1, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
<;
3
4
e8.nb
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@2D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 81, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
881, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@3D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 1, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, - 1, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@4D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 1, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, - 1, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@5D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 1, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, - 1, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@6D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 1, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, - 1, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@7D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 1, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, - 1, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@8D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 1, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, - 1, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@9D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 1<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, - 1<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@10D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 81, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
881, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@11D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 1, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, - 1, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@12D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 1, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, - 1, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@13D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 1, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, - 1, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@14D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 1, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, - 1, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@15D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 1, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, - 1, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@16D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 1, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, - 1, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@17D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 1<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, - 1<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@18D =
8 881  Sqrt@3D, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 81  Sqrt@3D, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<,
8- 2  Sqrt@3D, 0, 0, 0, 0, 0, 0, 0<<<;
B@19D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
,
e8.nb
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 81, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 81, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@20D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 1, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, - 1, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@21D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 1, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, - 1, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@22D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 1, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, - 1, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@23D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 1, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, - 1, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@24D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 1, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, - 1, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@25D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 1, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, - 1, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<<;
B@26D = 8 880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 1<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, - 1<, 80, 0, 0, 0, 0, 0, 0, 0<<<; B@27D =
8 88Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 8Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 8Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<<<;
AA = 8 88a@1D + a@18D  Sqrt@3D + a@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<,
8a@2D, a@3D, a@4D, a@5D, a@6D, a@7D, a@8D, a@9D<,
8a@10D, a@11D, a@12D, a@13D, a@14D, a@15D, a@16D, a@17D<<,
88a@2D, - a@3D, - a@4D, - a@5D, - a@6D, - a@7D, - a@8D, - a@9D<,
8- a@1D + a@18D  Sqrt@3D + a@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<,
8a@19D, a@20D, a@21D, a@22D, a@23D, a@24D, a@25D, a@26D<<,
88a@10D, - a@11D, - a@12D, - a@13D, - a@14D, - a@15D, - a@16D, - a@17D<,
8a@19D, - a@20D, - a@21D, - a@22D, - a@23D, - a@24D, - a@25D, - a@26D<,
8- 2 * a@18D  Sqrt@3D + a@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<<<;
BB = 8 88b@1D + b@18D  Sqrt@3D + b@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<,
8b@2D, b@3D, b@4D, b@5D, b@6D, b@7D, b@8D, b@9D<,
8b@10D, b@11D, b@12D, b@13D, b@14D, b@15D, b@16D, b@17D<<,
88b@2D, - b@3D, - b@4D, - b@5D, - b@6D, - b@7D, - b@8D, - b@9D<,
8- b@1D + b@18D  Sqrt@3D + b@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<,
8b@19D, b@20D, b@21D, b@22D, b@23D, b@24D, b@25D, b@26D<<,
88b@10D, - b@11D, - b@12D, - b@13D, - b@14D, - b@15D, - b@16D, - b@17D<,
8b@19D, - b@20D, - b@21D, - b@22D, - b@23D, - b@24D, - b@25D, - b@26D<,
8- 2 * b@18D  Sqrt@3D + b@27D * Sqrt@2  3D, 0, 0, 0, 0, 0, 0, 0<<<;
5
6
e8.nb
FF@AA_D := 8HPart@8Part@Part@Part@AA, 1D, 1D, 1D<, 1D Part@8Part@Part@Part@AA, 2D, 2D, 1D<, 1DL  2,
Part@8Part@Part@Part@AA, 1D, 2D, 1D<, 1D, Part@8Part@Part@Part@AA, 1D, 2D, 2D<, 1D,
Part@8Part@Part@Part@AA, 1D, 2D, 3D<, 1D, Part@8Part@Part@Part@AA, 1D, 2D, 4D<, 1D,
Part@8Part@Part@Part@AA, 1D, 2D, 5D<, 1D, Part@8Part@Part@Part@AA, 1D, 2D, 6D<, 1D,
Part@8Part@Part@Part@AA, 1D, 2D, 7D<, 1D, Part@8Part@Part@Part@AA, 1D, 2D, 8D<, 1D,
Part@8Part@Part@Part@AA, 1D, 3D, 1D<, 1D, Part@8Part@Part@Part@AA, 1D, 3D, 2D<, 1D,
Part@8Part@Part@Part@AA, 1D, 3D, 3D<, 1D, Part@8Part@Part@Part@AA, 1D, 3D, 4D<, 1D,
Part@8Part@Part@Part@AA, 1D, 3D, 5D<, 1D, Part@8Part@Part@Part@AA, 1D, 3D, 6D<, 1D,
Part@8Part@Part@Part@AA, 1D, 3D, 7D<, 1D,
Part@8Part@Part@Part@AA, 1D, 3D, 8D<, 1D,
Sqrt@3D  6 *
HPart@8Part@Part@Part@AA, 1D, 1D, 1D<, 1D + Part@8Part@Part@Part@AA, 2D, 2D, 1D<,
1D - 2 * Part@8Part@Part@Part@AA, 3D, 3D, 1D<, 1DL,
Part@8Part@Part@Part@AA, 2D, 3D, 1D<, 1D, Part@8Part@Part@Part@AA, 2D, 3D, 2D<, 1D,
Part@8Part@Part@Part@AA, 2D, 3D, 3D<, 1D,
Part@8Part@Part@Part@AA, 2D, 3D, 4D<, 1D,
Part@8Part@Part@Part@AA, 2D, 3D, 5D<, 1D,
Part@8Part@Part@Part@AA, 2D, 3D, 6D<, 1D,
Part@8Part@Part@Part@AA, 2D, 3D, 7D<, 1D, Part@8Part@Part@Part@AA, 2D, 3D, 8D<, 1D,
Sqrt@3  2D  3 * HPart@8Part@Part@Part@AA, 1D, 1D, 1D<, 1D + Part@
8Part@Part@Part@AA, 2D, 2D, 1D<, 1D + Part@8Part@Part@Part@AA, 3D, 3D, 1D<, 1DL<
FFi@SS_D := Sum@SS@@iDD * B@iD, 8i, 27<D;
basj = 8<;
Do@basj = basj ~ Join ~ 8D@MT, mt@iDD<, 8i, 26<D
Qm2@x_, y_D := Sum@Sum@x@@iDD y@@jDD QQ2@i, jD, 8i, 8<D, 8j, 8<D
QP2@x_, y_D := 8Coefficient@Qm2@x, yD, e@1DD, Coefficient@Qm2@x, yD, e@2DD,
Coefficient@Qm2@x, yD, e@3DD, Coefficient@Qm2@x, yD, e@4DD,
Coefficient@Qm2@x, yD, e@5DD, Coefficient@Qm2@x, yD, e@6DD,
Coefficient@Qm2@x, yD, e@7DD, Coefficient@Qm2@x, yD, e@8DD<
OctP@a_, b_D := 88Sum@QP2@Part@Part@a, 1D, iD, Part@Part@b, iD, 1DD, 8i, 3<D,
Sum@QP2@Part@Part@a, 1D, iD, Part@Part@b, iD, 2DD, 8i, 3<D,
Sum@QP2@Part@Part@a, 1D, iD, Part@Part@b, iD, 3DD, 8i, 3<D<,
8Sum@QP2@Part@Part@a, 2D, iD, Part@Part@b, iD, 1DD, 8i, 3<D,
Sum@QP2@Part@Part@a, 2D, iD, Part@Part@b, iD, 2DD, 8i, 3<D,
Sum@QP2@Part@Part@a, 2D, iD, Part@Part@b, iD, 3DD, 8i, 3<D<,
8Sum@QP2@Part@Part@a, 3D, iD, Part@Part@b, iD, 1DD, 8i, 3<D,
Sum@QP2@Part@Part@a, 3D, iD, Part@Part@b, iD, 2DD, 8i, 3<D,
Sum@QP2@Part@Part@a, 3D, iD, Part@Part@b, iD, 3DD, 8i, 3<D<<;
OctPS@a_, b_D := 1  2 HOctP@a, bD + OctP@b, aDL;
tracciamoct@a_D := Sum@a@@i, iDD, 8i, 3<D;
tracciamoctr@a_D := Sum@a@@i, iDD@@1DD, 8i, 3<D;
pscalj@a_, b_D := 1  2 * tracciamoctr@OctPS@a, bDD;
prod@a_, b_D :=
Simplify@OctPS@a, bD - 1  2 * tracciamoctr@aD * b - 1  2 * tracciamoctr@bD * a +
1  2 * Htracciamoctr@aD * tracciamoctr@bD - tracciamoctr@OctPS@a, bDDL *
B@27D * Sqrt@3  2DD;
e8.nb
7
MM = Array@mm, 827, 27<D;
vaa = FF@AAD; vbb = FF@BBD;
v1aa = MM.vaa;
v1bb = MM.vbb;
AA1 = FFi@v1aaD;
BB1 = FFi@v1bbD;
V1 = FF@OctPS@AA1, BBDD;
V2 = FF@OctPS@AA, BB1DD;
AB = OctPS@AA, BBD;
V = FF@ABD;
VV = MM.V;
diff = VV - V1 - V2;
Do@Do@Do@ff@i, j, kD = Coefficient@Part@diff, kD, a@iD b@jDD, 8i, 27<D, 8j, 27<D,
8k, 27<D;
n = 0;
Do@Do@Do@n ++;
If@ff@i, j, kD Š 0, n = n - 1, Ff@nD = ff@i, j, kD Š 0, Ff@nD = ff@i, j, kD Š 0D, 8i, 27<D,
8j, 27<D, 8k, 27<D
n
s@1D = 8<;
Do@s@i + 1D = Append@s@iD, Ff@iDD, 8i, n<D
m = 0;
Do@Do@8m ++, gg@mD = mm@i, jD<, 8i, 27<D, 8j, 27<D
v@1D = 8<;
Do@v@i + 1D = Append@v@iD, gg@iDD, 8i, 729<D
sol = Solve@s@nD, v@730DD
Do@Do@mat@i, jD = mm@i, jD, 8i, 27<D, 8j, 27<D;
Do@Do@Do@If@Part@Part@Part@sol, 1, i, 1DDD == mm@j, kD, mat@j, kD =
mm@j, kD . Part@Part@sol, 1, iDDD, 8i, 677<D, 8j, 27<D, 8k, 27<D
MM = Array@mat, 827, 27<D;
n = 0;
Do@Do@n ++; If@D@MM, mm@i, jDD Š DiagonalMatrix@
80, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0<D,
n = n - 1, Md@nD = D@MM, mm@i, jDD, Md@nD = D@MM, mm@i, jDDD, 8i, 27<D, 8j, 27<D;
n
H* Gram-Schmidt orthonormalization of the F4 generators *L
Mf4@1D = Md@1D;
Do@Do@
EE@j, iD = 0, 8i, 52<D, 8j, 52<D
Do@Do@EE@j, iD = - Tr@[email protected]@iDD  Tr@[email protected]@iDD, 8i, j - 1<D;
Mf4@jD = Md@jD + Sum@EE@j, iD Mf4@iD, 8i, j - 1<D, 8j, 2, 52<D
8
e8.nb
f4 = ConstantArray@0, 852, 27, 27<D;
Do@f4@@iDD = Mf4@iD  Sqrt@Abs@Tr@[email protected]@iDDDD, 8i, 52<D;
H* Structure Constants of F4 *L
Cf4 = ConstantArray@0, 852, 52, 52<D;
Do@Print@kD;
Do@Do@
Cf4@@i, j, kDD = Tr@Co@f4@@iDD, f4@@jDDD.f4@@kDDD  Tr@f4@@kDD.f4@@kDDD, 8i, 52<D,
8j, 52<D, 8k, 52<D
CO = ConstantArray@0, 87, 7, 14<D;
Do@CO@@i, j, kDD = Tr@OD@@i, jDD.g2@@kDDD  Tr@g2@@kDD.g2@@kDDD,
8i, 7<, 8j, 7<, 8k, 14<D;
h = Array@hh, 27D;
MM = Array@mm, 827, 27<D;
imm = FF@OctPS@MT, FFi@hDDD;
Do@Do@mm@i, jD = Coefficient@Part@imm, iD, Part@h, jDD, 8i, 27<D, 8j, 27<D
Do@Meconi@iD = D@MM, mt@iDD, 8i, 26<D
Me6@1D = Meconi@1D;
Do@Do@AAA@j, iD = 0, 8i, 26<D, 8j, 26<D
Do@Do@AAA@j, iD = - Tr@[email protected]@iDD  Tr@[email protected]@iDD, 8i, j - 1<D;
Me6@jD = Meconi@jD + Sum@AAA@j, iD Me6@iD, 8i, j - 1<D, 8j, 2, 26<D;
Do@Mne6@iD = Me6@iD  Sqrt@Abs@Tr@[email protected]@iDDDD, 8i, 26<D;
H* Structure constants of the matrices in E6 which are not in F4 *L
Ce6 = ConstantArray@0, 826, 26, 52<D;
Do@Print@kD;
Do@Do@
Ce6@@i, j, kDD = Tr@Co@Mne6@iD, [email protected]@@kDDD  Tr@f4@@kDD.f4@@kDDD, 8i, 26<D,
8j, 26<D, 8k, 52<D
H* Ingredients for Lie product in Tits formula *L
Do@s@i, j, kD = Comm@ee@i + 1D, ee@j + 1DD.ee@k + 1D, 8i, 7<, 8j, 7<, 8k, 7<D
BI = 8 881, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 81, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<<,
880, 0, 0, 0, 0, 0, 0, 0<, 80, 0, 0, 0, 0, 0, 0, 0<, 81, 0, 0, 0, 0, 0, 0, 0<<<;
KK = ConstantArray@0, 826, 26, 26<D;
Do@Print@kD;
Do@Do@KK@@i, j, kDD = tracciamoctr@
OctPS@OctPS@B@iD, B@jDD - H1  3L tracciamoctr@OctPS@B@iD, B@jDDD * BI, B@kDDD,
8i, j<D, 8j, k<D, 8k, 26<D;
KK2@8i_, j_, k_<D := KK@@i, j, kDD;
H* Adjoint of E8 *L
e8.nb
H* Map between entries in Tits formula and numbering of 248 matrices of E8 *L
R = ConstantArray@0, 8248, 4<D;
Do@R@@iDD = 8i, 0, 0, 0<, 8i, 14<D;
Do@R@@i + 14DD = 80, i, 0, 0<, 8i, 52<D;
Do@R@@i + 66DD =
80, 0, IntegerPart@Hi - 1L  26D + 1, i - IntegerPart@Hi - 1L  26D * 26<, 8i, 1, 182<D;
LL1 = ConstantArray@0, 7D;
Do@
LL1@@llDD = - Part@QP@ee@ll + 1D, ee@ll + 1DD, 1D, 8ll, 7<D
LL2 = ConstantArray@0, 26D;
Do@
LL2@@ADD = tracciamoctr@OctPS@B@AD, B@ADDD  2, 8A, 26<D
M = ConstantArray@0, 8248, 248, 248<D;
DoBM@@Flatten@Position@R, 8i1, 0, 0, 0<DD,
Flatten@Position@R, 8i2, 0, 0, 0<DD, Flatten@Position@R, 8i3, 0, 0, 0<DDDD =
1
Cg2@@i1, i3, i2DD , 8i1, 14<, 8i2, 14<, 8i3, 14<F;
3
Do@M@@Flatten@Position@R, 80, a1, 0, 0<DD,
Flatten@Position@R, 80, a2, 0, 0<DD, Flatten@Position@R, 80, a3, 0, 0<DDDD =
Cf4@@a1, a3, a2DD , 8a1, 52<, 8a2, 52<, 8a3, 52<D;
DoBM@@Flatten@Position@R, 8i1, 0, 0, 0<DD,
Flatten@Position@R, 80, 0, ll2, A2<DD, Flatten@Position@R, 80, 0, ll3, A2<DDDD =
1
g2@@i1, ll2, ll3DD , 8i1, 14<, 8ll2, 7<, 8ll3, 7<, 8A2, 26<F;
3
DoBM@@Flatten@Position@R, 80, 0, ll1, A1<DD, Flatten@Position@R, 8i2, 0, 0, 0<DD,
Flatten@Position@R, 80, 0, ll3, A1<DDDD =
1
LL2@@A1DD CO@@ll1, ll3, i2DD , 8i2, 14<, 8ll1, 7<, 8ll3, 7<, 8A1, 26<F;
12 3
Do@M@@Flatten@Position@R, 80, a1, 0, 0<DD,
Flatten@Position@R, 80, 0, ll2, A2<DD, Flatten@Position@R, 80, 0, ll2, A3<DDDD =
f4@@a1, A2, A3DD, 8a1, 52<, 8ll2, 7<, 8A2, 26<, 8A3, 26<D;
Do@M@@Flatten@Position@R, 80, 0, ll1, A1<DD, Flatten@Position@R, 80, a2, 0, 0<DD,
Flatten@Position@R, 80, 0, ll1, A3<DDDD =
- LL1@@ll1DD Ce6@@A1, A3, a2DD, 8a2, 52<, 8ll1, 7<, 8A1, 26<, 8A3, 26<D;
DoBM@@Flatten@Position@R, 80, 0, ll1, A1<DD,
Flatten@Position@R, 80, 0, ll2, A1<DD, Flatten@Position@R, 8i3, 0, 0, 0<DDDD =
1
g2@@i3, ll2, ll1DD, 8A1, 26<, 8ll1, 7<, 8ll2, 7<, 8i3, 14<F;
3
Do@M@@Flatten@Position@R, 80, 0, ll1, A1<DD,
Flatten@Position@R, 80, 0, ll1, A2<DD, Flatten@Position@R, 80, a3, 0, 0<DDDD =
- f4@@a3, A2, A1DD, 8A1, 26<, 8A2, 26<, 8a3, 52<, 8ll1, 7<D;
9
10
e8.nb
DoBDoBM@@Flatten@Position@R, 80, 0, ll1, A1<DD,
Flatten@Position@R, 80, 0, ll2, A2<DD, Flatten@Position@R, 80, 0, ll3, A3<DDDD =
1
s@ll1, ll3, ll2D KK2@Sort@8A1, A2, A3<DD,
4 6 LL2@@A2DD
8A1, 26<, 8A2, 26<, 8ll1, 7<, 8ll2, 7<, 8ll3, 7<F, 8A3, 26<F;