Root subsystems of E^{1}

$E^{1}_7$
Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.

Page generated by the calculator project.

g: E^{1}_7. There are 47 table entries (= 45 larger than the Cartan subalgebra + the Cartan subalgebra + the full subalgebra).

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

There are 32 parabolic, 12 pseudo-parabolic but not parabolic and 3 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", [63]],
["parabolic","2A^{1}_1", [63, 49]],
["parabolic","A^{1}_2", [63, 1]],
["parabolic","3A^{1}_1", [63, 49, 28]],
["parabolic","3A^{1}_1", [63, 49, 7]],
["parabolic","A^{1}_2+A^{1}_1", [63, 1, 30]],
["parabolic","A^{1}_3", [63, 1, 3]],
["parabolic","4A^{1}_1", [63, 49, 7, 28]],
["parabolic","A^{1}_2+2A^{1}_1", [63, 1, 30, 18]],
["parabolic","2A^{1}_2", [63, 1, 30, 7]],
["parabolic","A^{1}_3+A^{1}_1", [63, 1, 3, 19]],
["parabolic","A^{1}_3+A^{1}_1", [63, 1, 3, 2]],
["parabolic","A^{1}_4", [63, 1, 3, 4]],
["parabolic","D^{1}_4", [63, 1, 3, 28]],
["parabolic","A^{1}_2+3A^{1}_1", [63, 1, 30, 18, 5]],
["parabolic","2A^{1}_2+A^{1}_1", [63, 1, 30, 7, 11]],
["parabolic","A^{1}_3+2A^{1}_1", [63, 1, 3, 2, 19]],
["parabolic","A^{1}_3+A^{1}_2", [63, 1, 3, 19, 7]],
["parabolic","A^{1}_4+A^{1}_1", [63, 1, 3, 4, 13]],
["parabolic","D^{1}_4+A^{1}_1", [63, 1, 3, 28, 2]],
["parabolic","A^{1}_5", [63, 1, 3, 4, 5]],
["parabolic","A^{1}_5", [63, 1, 3, 4, 2]],
["parabolic","D^{1}_5", [63, 1, 3, 4, 16]],
["parabolic","A^{1}_3+A^{1}_2+A^{1}_1", [63, 1, 3, 19, 7, 2]],
["parabolic","A^{1}_4+A^{1}_2", [63, 1, 3, 4, 13, 7]],
["parabolic","A^{1}_5+A^{1}_1", [63, 1, 3, 4, 2, 13]],
["parabolic","D^{1}_5+A^{1}_1", [63, 1, 3, 4, 16, 7]],
["parabolic","A^{1}_6", [63, 1, 3, 4, 5, 6]],
["parabolic","D^{1}_6", [63, 1, 3, 4, 5, 2]],
["parabolic","E^{1}_6", [63, 16, 1, 3, 4, 12]],
["parabolic","E^{1}_7", [63, 16, 1, 3, 4, 12, 7]],
["pseudoParabolicNonParabolic","4A^{1}_1", [63, 49, 28, 3]],
["pseudoParabolicNonParabolic","5A^{1}_1", [63, 49, 7, 28, 2]],
["pseudoParabolicNonParabolic","A^{1}_3+2A^{1}_1", [63, 1, 3, 19, 6]],
["pseudoParabolicNonParabolic","3A^{1}_2", [63, 1, 30, 7, 11, 5]],
["pseudoParabolicNonParabolic","A^{1}_3+3A^{1}_1", [63, 1, 3, 2, 19, 6]],
["pseudoParabolicNonParabolic","2A^{1}_3", [63, 1, 3, 19, 7, 6]],
["pseudoParabolicNonParabolic","D^{1}_4+2A^{1}_1", [63, 1, 3, 28, 2, 5]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_1", [63, 1, 3, 4, 5, 7]],
["pseudoParabolicNonParabolic","2A^{1}_3+A^{1}_1", [63, 1, 3, 19, 7, 6, 2]],
["pseudoParabolicNonParabolic","A^{1}_5+A^{1}_2", [63, 1, 3, 4, 2, 13, 7]],
["pseudoParabolicNonParabolic","D^{1}_6+A^{1}_1", [63, 1, 3, 4, 5, 2, 7]],
["pseudoParabolicNonParabolic","A^{1}_7", [63, 1, 3, 4, 5, 6, 7]],
["nonPseudoParabolic","6A^{1}_1", [63, 49, 7, 28, 2, 3]],
["nonPseudoParabolic","7A^{1}_1", [63, 49, 7, 28, 2, 3, 5]],
["nonPseudoParabolic","D^{1}_4+3A^{1}_1", [63, 1, 3, 28, 2, 5, 7]]

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

LaTeX table with root subalgebra details.
\documentclass{article}
\usepackage{longtable, amssymb, lscape}
\begin{document}
Lie algebra type: $E^{1}_7$. There are 47 table entries (= 45 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
$E^{1}_7$&$0$&$63$&$0$&$0$&$7$&$0$&\\
$A^{1}_7$&$0$&$28$&$0$&$0$&$7$&$0$&\\
$D^{1}_6+A^{1}_1$&$0$&$31$&$0$&$1$&$7$&$0$&\\
$A^{1}_5+A^{1}_2$&$0$&$18$&$0$&$0$&$7$&$0$&\\
$D^{1}_4+3A^{1}_1$&$0$&$15$&$0$&$3$&$7$&$0$&\\
$2A^{1}_3+A^{1}_1$&$0$&$13$&$0$&$1$&$7$&$0$&\\
$7A^{1}_1$&$0$&$7$&$0$&$7$&$7$&$0$&\\
$E^{1}_6$&$0$&$36$&$0$&$0$&$6$&$0$&\\
$D^{1}_6$&$A^{1}_1$&$30$&$1$&$0$&$6$&$1$&\\
$A^{1}_6$&$0$&$21$&$0$&$0$&$6$&$0$&\\
$D^{1}_5+A^{1}_1$&$0$&$21$&$0$&$1$&$6$&$0$&\\
$A^{1}_5+A^{1}_1$&$0$&$16$&$0$&$1$&$6$&$0$&\\
$A^{1}_5+A^{1}_1$&$0$&$16$&$0$&$1$&$6$&$0$&\\
$D^{1}_4+2A^{1}_1$&$A^{1}_1$&$14$&$1$&$2$&$6$&$1$&\\
$A^{1}_4+A^{1}_2$&$0$&$13$&$0$&$0$&$6$&$0$&\\
$2A^{1}_3$&$A^{1}_1$&$12$&$1$&$0$&$6$&$1$&\\
$A^{1}_3+A^{1}_2+A^{1}_1$&$0$&$10$&$0$&$1$&$6$&$0$&\\
$A^{1}_3+3A^{1}_1$&$0$&$9$&$0$&$3$&$6$&$0$&\\
$3A^{1}_2$&$0$&$9$&$0$&$0$&$6$&$0$&\\
$6A^{1}_1$&$A^{1}_1$&$6$&$1$&$6$&$6$&$1$&\\
$D^{1}_5$&$A^{1}_1$&$20$&$1$&$0$&$5$&$1$&\\
$A^{1}_5$&$A^{1}_1$&$15$&$1$&$0$&$5$&$1$&\\
$A^{1}_5$&$A^{1}_2$&$15$&$3$&$0$&$5$&$2$&\\
$D^{1}_4+A^{1}_1$&$2A^{1}_1$&$13$&$2$&$1$&$5$&$2$&\\
$A^{1}_4+A^{1}_1$&$0$&$11$&$0$&$1$&$5$&$0$&\\
$A^{1}_3+A^{1}_2$&$A^{1}_1$&$9$&$1$&$0$&$5$&$1$&\\
$A^{1}_3+2A^{1}_1$&$A^{1}_1$&$8$&$1$&$2$&$5$&$1$&\\
$A^{1}_3+2A^{1}_1$&$A^{1}_1$&$8$&$1$&$2$&$5$&$1$&\\
$2A^{1}_2+A^{1}_1$&$0$&$7$&$0$&$1$&$5$&$0$&\\
$A^{1}_2+3A^{1}_1$&$0$&$6$&$0$&$3$&$5$&$0$&\\
$5A^{1}_1$&$2A^{1}_1$&$5$&$2$&$5$&$5$&$2$&\\
$D^{1}_4$&$3A^{1}_1$&$12$&$3$&$0$&$4$&$3$&\\
$A^{1}_4$&$A^{1}_2$&$10$&$3$&$0$&$4$&$2$&\\
$A^{1}_3+A^{1}_1$&$2A^{1}_1$&$7$&$2$&$1$&$4$&$2$&\\
$A^{1}_3+A^{1}_1$&$A^{1}_3$&$7$&$6$&$1$&$4$&$3$&\\
$2A^{1}_2$&$A^{1}_2$&$6$&$3$&$0$&$4$&$2$&\\
$A^{1}_2+2A^{1}_1$&$A^{1}_1$&$5$&$1$&$2$&$4$&$1$&\\
$4A^{1}_1$&$3A^{1}_1$&$4$&$3$&$4$&$4$&$3$&\\
$4A^{1}_1$&$3A^{1}_1$&$4$&$3$&$4$&$4$&$3$&\\
$A^{1}_3$&$A^{1}_3+A^{1}_1$&$6$&$7$&$0$&$3$&$4$&\\
$A^{1}_2+A^{1}_1$&$A^{1}_3$&$4$&$6$&$1$&$3$&$3$&\\
$3A^{1}_1$&$4A^{1}_1$&$3$&$4$&$3$&$3$&$4$&\\
$3A^{1}_1$&$D^{1}_4$&$3$&$12$&$3$&$3$&$4$&\\
$A^{1}_2$&$A^{1}_5$&$3$&$15$&$0$&$2$&$5$&\\
$2A^{1}_1$&$D^{1}_4+A^{1}_1$&$2$&$13$&$2$&$2$&$5$&\\
$A^{1}_1$&$D^{1}_6$&$1$&$30$&$1$&$1$&$6$&\\
$0$&$E^{1}_7$&$0$&$63$&$0$&$0$&$7$&\\
\end{longtable}
\end{document}