Table 6: The 9-dimensional candidates of nullities 6 and 8.

Lie bracket Index Nullity

1 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45) 6 6
2 (0, 0, 1.12, 1.13, 1.14, 1.15, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45) 7 6
3 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45) 7 6
4 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45) 7 6
5 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45) 7 6
6 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45) 7 6
7 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36) 7 6
8 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.36 + 1.45) 7 6
9 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.35, 1.18 + 1.27 + 1.36 + 1.45) 7 6
10 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45) 7 6
11 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45) 7 6
12 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36) 7 6
13 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.36 + 1.45) 7 6
14 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45) 7 6
15 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45) 7 6
16 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36) 7 6
17 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45) 7 6
18 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45) 7 6
19 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36) 7 6
20 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45) 7 6
21 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36) 7 6
22 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27) 7 6
23 (0, 0, 1.12, 1.13, 1.14 + 1.23, 1.15 + 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36 + 1.45) 7 8