Root subsystems of E^{1}

$E^{1}_8$
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.

Page generated by the calculator project.

g: E^{1}_8. There are 77 table entries (= 75 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).

Type k_{ss}: E^{1}_8 (Full subalgebra) Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_8 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_8 Type C(k_{ss})_{ss}: 0	Type k_{ss}: E^{1}_7+A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_7+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: E^{1}_6+A^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_6+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_5+A^{1}_3 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_5+A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 2D^{1}_4 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_4+4A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 2A^{1}_4 Type C(k_{ss})_{ss}: 0
Type k_{ss}: 2A^{1}_3+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 4A^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 8A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: E^{1}_7 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: D^{1}_7 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_7 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_7 Type C(k_{ss})_{ss}: 0	Type k_{ss}: E^{1}_6+A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: D^{1}_6+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_6+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_5+A^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_5+2A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_5+A^{1}_2 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_5+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_4+A^{1}_3 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_4+3A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: A^{1}_4+A^{1}_3 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_4+A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 2A^{1}_3+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_3+A^{1}_2+2A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: A^{1}_3+4A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 3A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 7A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: E^{1}_6 Type C(k_{ss})_{ss}: A^{1}_2
Type k_{ss}: D^{1}_6 Type C(k_{ss})_{ss}: 2A^{1}_1	Type k_{ss}: A^{1}_6 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: D^{1}_5+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: A^{1}_5+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: A^{1}_5+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_2	Type k_{ss}: D^{1}_4+A^{1}_2 Type C(k_{ss})_{ss}: 0	Type k_{ss}: D^{1}_4+2A^{1}_1 Type C(k_{ss})_{ss}: 2A^{1}_1	Type k_{ss}: A^{1}_4+A^{1}_2 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: A^{1}_4+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: 2A^{1}_3 Type C(k_{ss})_{ss}: 2A^{1}_1	Type k_{ss}: 2A^{1}_3 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_3+A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1
Type k_{ss}: A^{1}_3+3A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: 3A^{1}_2 Type C(k_{ss})_{ss}: A^{1}_2	Type k_{ss}: 2A^{1}_2+2A^{1}_1 Type C(k_{ss})_{ss}: 0	Type k_{ss}: A^{1}_2+4A^{1}_1 Type C(k_{ss})_{ss}: 0
Type k_{ss}: 6A^{1}_1 Type C(k_{ss})_{ss}: 2A^{1}_1	Type k_{ss}: D^{1}_5 Type C(k_{ss})_{ss}: A^{1}_3	Type k_{ss}: A^{1}_5 Type C(k_{ss})_{ss}: A^{1}_2+A^{1}_1	Type k_{ss}: D^{1}_4+A^{1}_1 Type C(k_{ss})_{ss}: 3A^{1}_1
Type k_{ss}: A^{1}_4+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_2	Type k_{ss}: A^{1}_3+A^{1}_2 Type C(k_{ss})_{ss}: 2A^{1}_1	Type k_{ss}: A^{1}_3+2A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_3	Type k_{ss}: A^{1}_3+2A^{1}_1 Type C(k_{ss})_{ss}: 2A^{1}_1
Type k_{ss}: 2A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_2	Type k_{ss}: A^{1}_2+3A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_1	Type k_{ss}: 5A^{1}_1 Type C(k_{ss})_{ss}: 3A^{1}_1	Type k_{ss}: D^{1}_4 Type C(k_{ss})_{ss}: D^{1}_4
Type k_{ss}: A^{1}_4 Type C(k_{ss})_{ss}: A^{1}_4	Type k_{ss}: A^{1}_3+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_3+A^{1}_1	Type k_{ss}: 2A^{1}_2 Type C(k_{ss})_{ss}: 2A^{1}_2	Type k_{ss}: A^{1}_2+2A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_3
Type k_{ss}: 4A^{1}_1 Type C(k_{ss})_{ss}: D^{1}_4	Type k_{ss}: 4A^{1}_1 Type C(k_{ss})_{ss}: 4A^{1}_1	Type k_{ss}: A^{1}_3 Type C(k_{ss})_{ss}: D^{1}_5	Type k_{ss}: A^{1}_2+A^{1}_1 Type C(k_{ss})_{ss}: A^{1}_5
Type k_{ss}: 3A^{1}_1 Type C(k_{ss})_{ss}: D^{1}_4+A^{1}_1	Type k_{ss}: A^{1}_2 Type C(k_{ss})_{ss}: E^{1}_6	Type k_{ss}: 2A^{1}_1 Type C(k_{ss})_{ss}: D^{1}_6	Type k_{ss}: A^{1}_1 Type C(k_{ss})_{ss}: E^{1}_7
Type k_{ss}: 0 (Cartan subalgebra) Type C(k_{ss})_{ss}: E^{1}_8

There are 41 parabolic, 26 pseudo-parabolic but not parabolic and 10 non pseudo-parabolic root subsystems. The roots needed to generate the root subsystems are listed below using the root indices in GAP.
["parabolic","0", []],
["parabolic","A^{1}_1", [120]],
["parabolic","2A^{1}_1", [120, 97]],
["parabolic","A^{1}_2", [120, 8]],
["parabolic","3A^{1}_1", [120, 97, 61]],
["parabolic","A^{1}_2+A^{1}_1", [120, 8, 69]],
["parabolic","A^{1}_3", [120, 8, 7]],
["parabolic","4A^{1}_1", [120, 97, 61, 32]],
["parabolic","A^{1}_2+2A^{1}_1", [120, 8, 69, 31]],
["parabolic","2A^{1}_2", [120, 8, 69, 2]],
["parabolic","A^{1}_3+A^{1}_1", [120, 8, 7, 44]],
["parabolic","A^{1}_4", [120, 8, 7, 6]],
["parabolic","D^{1}_4", [120, 8, 7, 61]],
["parabolic","A^{1}_2+3A^{1}_1", [120, 8, 69, 31, 19]],
["parabolic","2A^{1}_2+A^{1}_1", [120, 8, 69, 2, 9]],
["parabolic","A^{1}_3+2A^{1}_1", [120, 8, 7, 44, 18]],
["parabolic","A^{1}_3+A^{1}_2", [120, 8, 7, 44, 3]],
["parabolic","A^{1}_4+A^{1}_1", [120, 8, 7, 6, 23]],
["parabolic","D^{1}_4+A^{1}_1", [120, 8, 7, 61, 32]],
["parabolic","A^{1}_5", [120, 8, 7, 6, 5]],
["parabolic","D^{1}_5", [120, 8, 7, 6, 48]],
["parabolic","2A^{1}_2+2A^{1}_1", [120, 8, 69, 2, 9, 13]],
["parabolic","A^{1}_3+A^{1}_2+A^{1}_1", [120, 8, 7, 44, 3, 2]],
["parabolic","2A^{1}_3", [120, 8, 7, 44, 3, 4]],
["parabolic","A^{1}_4+2A^{1}_1", [120, 8, 7, 6, 23, 11]],
["parabolic","A^{1}_4+A^{1}_2", [120, 8, 7, 6, 23, 2]],
["parabolic","D^{1}_4+A^{1}_2", [120, 8, 7, 61, 32, 4]],
["parabolic","A^{1}_5+A^{1}_1", [120, 8, 7, 6, 5, 9]],
["parabolic","D^{1}_5+A^{1}_1", [120, 8, 7, 6, 48, 17]],
["parabolic","A^{1}_6", [120, 8, 7, 6, 5, 4]],
["parabolic","D^{1}_6", [120, 8, 7, 6, 5, 32]],
["parabolic","E^{1}_6", [120, 48, 8, 7, 6, 24]],
["parabolic","A^{1}_4+A^{1}_2+A^{1}_1", [120, 8, 7, 6, 23, 2, 3]],
["parabolic","A^{1}_4+A^{1}_3", [120, 8, 7, 6, 23, 2, 4]],
["parabolic","D^{1}_5+A^{1}_2", [120, 8, 7, 6, 48, 17, 3]],
["parabolic","A^{1}_6+A^{1}_1", [120, 8, 7, 6, 5, 4, 1]],
["parabolic","E^{1}_6+A^{1}_1", [120, 48, 8, 7, 6, 24, 11]],
["parabolic","A^{1}_7", [120, 8, 7, 6, 5, 4, 3]],
["parabolic","D^{1}_7", [120, 8, 7, 6, 5, 4, 17]],
["parabolic","E^{1}_7", [120, 48, 8, 7, 6, 24, 2]],
["parabolic","E^{1}_8", [120, 48, 8, 7, 6, 24, 2, 4]],
["pseudoParabolicNonParabolic","4A^{1}_1", [120, 97, 61, 7]],
["pseudoParabolicNonParabolic","5A^{1}_1", [120, 97, 61, 7, 32]],
["pseudoParabolicNonParabolic","A^{1}_3+2A^{1}_1", [120, 8, 7, 44, 1]],
["pseudoParabolicNonParabolic","A^{1}_2+4A^{1}_1", [120, 8, 69, 31, 19, 4]],
["pseudoParabolicNonParabolic","3A^{1}_2", [120, 8, 69, 2, 9, 3]],
["pseudoParabolicNonParabolic","A^{1}_3+3A^{1}_1", [120, 8, 7, 44, 1, 18]],
["pseudoParabolicNonParabolic","2A^{1}_3", [120, 8, 7, 44, 3, 1]],
["pseudoParabolicNonParabolic","D^{1}_4+2A^{1}_1", [120, 8, 7, 61, 32, 2]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_1", [120, 8, 7, 6, 5, 2]],
["pseudoParabolicNonParabolic","3A^{1}_2+A^{1}_1", [120, 8, 69, 2, 9, 3, 13]],
["pseudoParabolicNonParabolic","A^{1}_3+A^{1}_2+2A^{1}_1", [120, 8, 7, 44, 3, 2, 5]],
["pseudoParabolicNonParabolic","2A^{1}_3+A^{1}_1", [120, 8, 7, 44, 3, 1, 2]],
["pseudoParabolicNonParabolic","D^{1}_4+A^{1}_3", [120, 8, 7, 61, 32, 4, 5]],
["pseudoParabolicNonParabolic","A^{1}_5+2A^{1}_1", [120, 8, 7, 6, 5, 2, 9]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_2", [120, 8, 7, 6, 5, 9, 3]],
["pseudoParabolicNonParabolic","D^{1}_5+2A^{1}_1", [120, 8, 7, 6, 48, 17, 4]],
["pseudoParabolicNonParabolic","D^{1}_6+A^{1}_1", [120, 8, 7, 6, 5, 32, 2]],
["pseudoParabolicNonParabolic","A^{1}_7", [120, 8, 7, 6, 5, 4, 2]],
["pseudoParabolicNonParabolic","2A^{1}_4", [120, 8, 7, 6, 23, 2, 4, 3]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_2+A^{1}_1", [120, 8, 7, 6, 5, 9, 3, 2]],
["pseudoParabolicNonParabolic","D^{1}_5+A^{1}_3", [120, 8, 7, 6, 48, 17, 3, 4]],
["pseudoParabolicNonParabolic","E^{1}_6+A^{1}_2", [120, 48, 8, 7, 6, 24, 11, 4]],
["pseudoParabolicNonParabolic","A^{1}_7+A^{1}_1", [120, 8, 7, 6, 5, 4, 2, 1]],
["pseudoParabolicNonParabolic","E^{1}_7+A^{1}_1", [120, 48, 8, 7, 6, 24, 2, 3]],
["pseudoParabolicNonParabolic","A^{1}_8", [120, 8, 7, 6, 5, 4, 3, 1]],
["pseudoParabolicNonParabolic","D^{1}_8", [120, 8, 7, 6, 5, 4, 3, 2]],
["nonPseudoParabolic","6A^{1}_1", [120, 97, 61, 7, 32, 2]],
["nonPseudoParabolic","7A^{1}_1", [120, 97, 61, 7, 32, 2, 3]],
["nonPseudoParabolic","A^{1}_3+4A^{1}_1", [120, 8, 7, 44, 1, 18, 4]],
["nonPseudoParabolic","D^{1}_4+3A^{1}_1", [120, 8, 7, 61, 32, 2, 3]],
["nonPseudoParabolic","8A^{1}_1", [120, 97, 61, 7, 32, 2, 3, 5]],
["nonPseudoParabolic","4A^{1}_2", [120, 8, 69, 2, 9, 3, 13, 6]],
["nonPseudoParabolic","2A^{1}_3+2A^{1}_1", [120, 8, 7, 44, 3, 1, 2, 5]],
["nonPseudoParabolic","D^{1}_4+4A^{1}_1", [120, 8, 7, 61, 32, 2, 3, 5]],
["nonPseudoParabolic","2D^{1}_4", [120, 8, 7, 61, 32, 4, 5, 3]],
["nonPseudoParabolic","D^{1}_6+2A^{1}_1", [120, 8, 7, 6, 5, 32, 2, 3]]

The roots needed to generate the root subsystems are listed below.
["parabolic","0", []],
["parabolic","A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2]]],
["parabolic","2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0]]],
["parabolic","A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1]]],
["parabolic","3A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0], [0, 1, 1, 2, 2, 2, 1, 0]]],
["parabolic","A^{1}_2+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0]]],
["parabolic","A^{1}_3", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0]]],
["parabolic","4A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0], [0, 1, 1, 2, 2, 2, 1, 0], [0, 1, 1, 2, 1, 0, 0, 0]]],
["parabolic","A^{1}_2+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0]]],
["parabolic","2A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_3+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0]]],
["parabolic","A^{1}_4", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0]]],
["parabolic","D^{1}_4", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0]]],
["parabolic","A^{1}_2+3A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0]]],
["parabolic","2A^{1}_2+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_3+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 1, 0, 1, 1, 0, 0, 0]]],
["parabolic","A^{1}_3+A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_4+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 0]]],
["parabolic","D^{1}_4+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0], [0, 1, 1, 2, 1, 0, 0, 0]]],
["parabolic","A^{1}_5", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0]]],
["parabolic","D^{1}_5", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, -2, -2, -1, 0, 0]]],
["parabolic","2A^{1}_2+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0]]],
["parabolic","A^{1}_3+A^{1}_2+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["parabolic","2A^{1}_3", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","A^{1}_4+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0]]],
["parabolic","A^{1}_4+A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0]]],
["parabolic","D^{1}_4+A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","A^{1}_5+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0]]],
["parabolic","D^{1}_5+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, -2, -2, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0]]],
["parabolic","A^{1}_6", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","D^{1}_6", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, -1, -1, -2, -1, 0, 0, 0]]],
["parabolic","E^{1}_6", [[2, 3, 4, 6, 5, 4, 3, 2], [0, -1, -1, -2, -2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, -1, -1, 0, 0, 0]]],
["parabolic","A^{1}_4+A^{1}_2+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_4+A^{1}_3", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["parabolic","D^{1}_5+A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, -2, -2, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["parabolic","A^{1}_6+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]]],
["parabolic","E^{1}_6+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, -1, -1, -2, -2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, -1, -1, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0]]],
["parabolic","A^{1}_7", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["parabolic","D^{1}_7", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, -1, -1, -1, 0, 0, 0, 0]]],
["parabolic","E^{1}_7", [[2, 3, 4, 6, 5, 4, 3, 2], [0, -1, -1, -2, -2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, -1, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0]]],
["parabolic","E^{1}_8", [[2, 3, 4, 6, 5, 4, 3, 2], [0, -1, -1, -2, -2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, -1, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","4A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0], [0, 1, 1, 2, 2, 2, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0]]],
["pseudoParabolicNonParabolic","5A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0], [0, 1, 1, 2, 2, 2, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 1, 2, 1, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_3+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_2+4A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [1, 0, 1, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","3A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_3+3A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 1, 0, 0, 0]]],
["pseudoParabolicNonParabolic","2A^{1}_3", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","D^{1}_4+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","3A^{1}_2+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_3+A^{1}_2+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0]]],
["pseudoParabolicNonParabolic","2A^{1}_3+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","D^{1}_4+A^{1}_3", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_5+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","D^{1}_5+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, -2, -2, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","D^{1}_6+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, -1, -1, -2, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_7", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","2A^{1}_4", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [1, 1, 1, 1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_2+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","D^{1}_5+A^{1}_3", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, -1, -1, -2, -2, -1, 0, 0], [0, 1, 1, 1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","E^{1}_6+A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, -1, -1, -2, -2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, -1, -1, 0, 0, 0], [0, 0, 1, 1, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_7+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","E^{1}_7+A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, -1, -1, -2, -2, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [-1, 0, -1, -1, -1, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","A^{1}_8", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0]]],
["pseudoParabolicNonParabolic","D^{1}_8", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0]]],
["nonPseudoParabolic","6A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0], [0, 1, 1, 2, 2, 2, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0]]],
["nonPseudoParabolic","7A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0], [0, 1, 1, 2, 2, 2, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]],
["nonPseudoParabolic","A^{1}_3+4A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 1, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0, 0]]],
["nonPseudoParabolic","D^{1}_4+3A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]],
["nonPseudoParabolic","8A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [2, 2, 3, 4, 3, 2, 1, 0], [0, 1, 1, 2, 2, 2, 1, 0], [0, 0, 0, 0, 0, 0, 1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0]]],
["nonPseudoParabolic","4A^{1}_2", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [1, 2, 2, 3, 2, 1, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0], [1, 0, 1, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 1, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0]]],
["nonPseudoParabolic","2A^{1}_3+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [1, 1, 2, 2, 1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0], [-1, 0, 0, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0]]],
["nonPseudoParabolic","D^{1}_4+4A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 0, 0, 0]]],
["nonPseudoParabolic","2D^{1}_4", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, -1, -1, -2, -2, -2, -1, 0], [0, 1, 1, 2, 1, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0]]],
["nonPseudoParabolic","D^{1}_6+2A^{1}_1", [[2, 3, 4, 6, 5, 4, 3, 2], [0, 0, 0, 0, 0, 0, 0, -1], [0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0], [0, -1, -1, -2, -1, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0, 0], [0, 0, 1, 0, 0, 0, 0, 0]]]

LaTeX table with root subalgebra details.
\documentclass{article}
\usepackage{longtable, amssymb, lscape}
\begin{document}
Lie algebra type: $E^{1}_8$. There are 77 table entries (= 75 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).
Let $\mathfrak g$ stand for the type of the regular subalgebra and $C(\mathfrak g)$ for the type of the centralizer. Let $r$ stand for the rank of $\mathfrak g$, let $r_c$ stand for the rank of the semisimple part of the centralizer, let $p$ stand for the number of positive roots of $\mathfrak g$, let $q$ stand for the number of positive roots of the centralizer, and let $m$ stand for the number of $A_1$ components (of all root lengths) of $\mathfrak g$. \begin{longtable}{cccccccc}
$\mathfrak g$ & $C(\mathfrak g)$& $p$ & $q$& $m$& $r$ & $c_r$ \\\endhead
$E^{1}_8$&$0$&$120$&$0$&$0$&$8$&$0$&\\
$D^{1}_8$&$0$&$56$&$0$&$0$&$8$&$0$&\\
$A^{1}_8$&$0$&$36$&$0$&$0$&$8$&$0$&\\
$E^{1}_7+A^{1}_1$&$0$&$64$&$0$&$1$&$8$&$0$&\\
$A^{1}_7+A^{1}_1$&$0$&$29$&$0$&$1$&$8$&$0$&\\
$E^{1}_6+A^{1}_2$&$0$&$39$&$0$&$0$&$8$&$0$&\\
$D^{1}_6+2A^{1}_1$&$0$&$32$&$0$&$2$&$8$&$0$&\\
$D^{1}_5+A^{1}_3$&$0$&$26$&$0$&$0$&$8$&$0$&\\
$A^{1}_5+A^{1}_2+A^{1}_1$&$0$&$19$&$0$&$1$&$8$&$0$&\\
$2D^{1}_4$&$0$&$24$&$0$&$0$&$8$&$0$&\\
$D^{1}_4+4A^{1}_1$&$0$&$16$&$0$&$4$&$8$&$0$&\\
$2A^{1}_4$&$0$&$20$&$0$&$0$&$8$&$0$&\\
$2A^{1}_3+2A^{1}_1$&$0$&$14$&$0$&$2$&$8$&$0$&\\
$4A^{1}_2$&$0$&$12$&$0$&$0$&$8$&$0$&\\
$8A^{1}_1$&$0$&$8$&$0$&$8$&$8$&$0$&\\
$E^{1}_7$&$A^{1}_1$&$63$&$1$&$0$&$7$&$1$&\\
$D^{1}_7$&$0$&$42$&$0$&$0$&$7$&$0$&\\
$A^{1}_7$&$A^{1}_1$&$28$&$1$&$0$&$7$&$1$&\\
$A^{1}_7$&$0$&$28$&$0$&$0$&$7$&$0$&\\
$E^{1}_6+A^{1}_1$&$0$&$37$&$0$&$1$&$7$&$0$&\\
$D^{1}_6+A^{1}_1$&$A^{1}_1$&$31$&$1$&$1$&$7$&$1$&\\
$A^{1}_6+A^{1}_1$&$0$&$22$&$0$&$1$&$7$&$0$&\\
$D^{1}_5+A^{1}_2$&$0$&$23$&$0$&$0$&$7$&$0$&\\
$D^{1}_5+2A^{1}_1$&$0$&$22$&$0$&$2$&$7$&$0$&\\
$A^{1}_5+A^{1}_2$&$A^{1}_1$&$18$&$1$&$0$&$7$&$1$&\\
$A^{1}_5+2A^{1}_1$&$0$&$17$&$0$&$2$&$7$&$0$&\\
$D^{1}_4+A^{1}_3$&$0$&$18$&$0$&$0$&$7$&$0$&\\
$D^{1}_4+3A^{1}_1$&$A^{1}_1$&$15$&$1$&$3$&$7$&$1$&\\
$A^{1}_4+A^{1}_3$&$0$&$16$&$0$&$0$&$7$&$0$&\\
$A^{1}_4+A^{1}_2+A^{1}_1$&$0$&$14$&$0$&$1$&$7$&$0$&\\
$2A^{1}_3+A^{1}_1$&$A^{1}_1$&$13$&$1$&$1$&$7$&$1$&\\
$A^{1}_3+A^{1}_2+2A^{1}_1$&$0$&$11$&$0$&$2$&$7$&$0$&\\
$A^{1}_3+4A^{1}_1$&$0$&$10$&$0$&$4$&$7$&$0$&\\
$3A^{1}_2+A^{1}_1$&$0$&$10$&$0$&$1$&$7$&$0$&\\
$7A^{1}_1$&$A^{1}_1$&$7$&$1$&$7$&$7$&$1$&\\
$E^{1}_6$&$A^{1}_2$&$36$&$3$&$0$&$6$&$2$&\\
$D^{1}_6$&$2A^{1}_1$&$30$&$2$&$0$&$6$&$2$&\\
$A^{1}_6$&$A^{1}_1$&$21$&$1$&$0$&$6$&$1$&\\
$D^{1}_5+A^{1}_1$&$A^{1}_1$&$21$&$1$&$1$&$6$&$1$&\\
$A^{1}_5+A^{1}_1$&$A^{1}_1$&$16$&$1$&$1$&$6$&$1$&\\
$A^{1}_5+A^{1}_1$&$A^{1}_2$&$16$&$3$&$1$&$6$&$2$&\\
$D^{1}_4+A^{1}_2$&$0$&$15$&$0$&$0$&$6$&$0$&\\
$D^{1}_4+2A^{1}_1$&$2A^{1}_1$&$14$&$2$&$2$&$6$&$2$&\\
$A^{1}_4+A^{1}_2$&$A^{1}_1$&$13$&$1$&$0$&$6$&$1$&\\
$A^{1}_4+2A^{1}_1$&$0$&$12$&$0$&$2$&$6$&$0$&\\
$2A^{1}_3$&$2A^{1}_1$&$12$&$2$&$0$&$6$&$2$&\\
$2A^{1}_3$&$0$&$12$&$0$&$0$&$6$&$0$&\\
$A^{1}_3+A^{1}_2+A^{1}_1$&$A^{1}_1$&$10$&$1$&$1$&$6$&$1$&\\
$A^{1}_3+3A^{1}_1$&$A^{1}_1$&$9$&$1$&$3$&$6$&$1$&\\
$3A^{1}_2$&$A^{1}_2$&$9$&$3$&$0$&$6$&$2$&\\
$2A^{1}_2+2A^{1}_1$&$0$&$8$&$0$&$2$&$6$&$0$&\\
$A^{1}_2+4A^{1}_1$&$0$&$7$&$0$&$4$&$6$&$0$&\\
$6A^{1}_1$&$2A^{1}_1$&$6$&$2$&$6$&$6$&$2$&\\
$D^{1}_5$&$A^{1}_3$&$20$&$6$&$0$&$5$&$3$&\\
$A^{1}_5$&$A^{1}_2+A^{1}_1$&$15$&$4$&$0$&$5$&$3$&\\
$D^{1}_4+A^{1}_1$&$3A^{1}_1$&$13$&$3$&$1$&$5$&$3$&\\
$A^{1}_4+A^{1}_1$&$A^{1}_2$&$11$&$3$&$1$&$5$&$2$&\\
$A^{1}_3+A^{1}_2$&$2A^{1}_1$&$9$&$2$&$0$&$5$&$2$&\\
$A^{1}_3+2A^{1}_1$&$A^{1}_3$&$8$&$6$&$2$&$5$&$3$&\\
$A^{1}_3+2A^{1}_1$&$2A^{1}_1$&$8$&$2$&$2$&$5$&$2$&\\
$2A^{1}_2+A^{1}_1$&$A^{1}_2$&$7$&$3$&$1$&$5$&$2$&\\
$A^{1}_2+3A^{1}_1$&$A^{1}_1$&$6$&$1$&$3$&$5$&$1$&\\
$5A^{1}_1$&$3A^{1}_1$&$5$&$3$&$5$&$5$&$3$&\\
$D^{1}_4$&$D^{1}_4$&$12$&$12$&$0$&$4$&$4$&\\
$A^{1}_4$&$A^{1}_4$&$10$&$10$&$0$&$4$&$4$&\\
$A^{1}_3+A^{1}_1$&$A^{1}_3+A^{1}_1$&$7$&$7$&$1$&$4$&$4$&\\
$2A^{1}_2$&$2A^{1}_2$&$6$&$6$&$0$&$4$&$4$&\\
$A^{1}_2+2A^{1}_1$&$A^{1}_3$&$5$&$6$&$2$&$4$&$3$&\\
$4A^{1}_1$&$D^{1}_4$&$4$&$12$&$4$&$4$&$4$&\\
$4A^{1}_1$&$4A^{1}_1$&$4$&$4$&$4$&$4$&$4$&\\
$A^{1}_3$&$D^{1}_5$&$6$&$20$&$0$&$3$&$5$&\\
$A^{1}_2+A^{1}_1$&$A^{1}_5$&$4$&$15$&$1$&$3$&$5$&\\
$3A^{1}_1$&$D^{1}_4+A^{1}_1$&$3$&$13$&$3$&$3$&$5$&\\
$A^{1}_2$&$E^{1}_6$&$3$&$36$&$0$&$2$&$6$&\\
$2A^{1}_1$&$D^{1}_6$&$2$&$30$&$2$&$2$&$6$&\\
$A^{1}_1$&$E^{1}_7$&$1$&$63$&$1$&$1$&$7$&\\
$0$&$E^{1}_8$&$0$&$120$&$0$&$0$&$8$&\\
\end{longtable}
\end{document}