Table 7: The 9-dimensional nilsoliton metric Lie algebra candidates of nullity 3.

Lie bracket Index Nullity

1 (0, 0, 0, 0, 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18, 1.27 + 1.36 + 1.45) 4 3
2 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45) 5 3
3 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45) 5 3
4 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36) 5 3
5 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45) 5 3
6 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45) 5 3
7 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36) 5 3
8 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45) 5 3
9 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36) 5 3
10 (0, 0, 1.12, 0, 1.23, 1.24, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27) 5 3
11 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45) 5 3
12 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45) 5 3
13 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36) 5 3
14 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45) 5 3
15 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36) 5 3
16 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45) 5 3
17 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45) 5 3
18 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45) 5 3
19 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45) 5 3
20 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36) 5 3
21 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27) 5 3
22 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45) 5 3
23 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45) 5 3
24 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36) 5 3
25 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45) 5 3
26 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45) 5 3
27 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45) 5 3
28 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36) 5 3
29 (0, 0, 1.12, 0, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27) 5 3
30 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.16 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45) 4 3
31 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.16 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36) 4 3
32 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45) 4 3
33 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.36) 4 3
34 (0, 0, 1.12, 0, 1.14 + 1.23, 1.24, 1.16 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27) 4 3
35 (0, 0, 1.12, 1.13, 1.23, 0, 1.16 + 1.25, 1.17 + 1.26 + 1.35, 1.27 + 1.36 + 1.45) 4 3
36 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.26 + 1.35, 1.18 + 1.27 + 1.45) 5 3
37 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.26 + 1.35, 1.18 + 1.27 + 1.36) 5 3
38 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26, 1.27 + 1.36 + 1.45) 5 3
39 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.27 + 1.45) 5 3
40 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.27 + 1.36) 5 3
41 (0, 0, 1.12, 1.13, 1.14, 0, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36 + 1.45) 5 3
42 (0, 0, 1.12, 1.13, 1.14, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.45) 6 3
43 (0, 0, 1.12, 1.13, 1.14, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27 + 1.36) 6 3
44 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.36 + 1.45) 6 3
45 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.45) 6 3
46 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45) 6 3
47 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36) 6 3
48 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27) 6 3
49 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.26 + 1.35, 1.27 + 1.45) 5 3
50 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.26 + 1.35, 1.27 + 1.36) 5 3
51 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.35, 1.18 + 1.27 + 1.45) 5 3
52 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.35, 1.18 + 1.27 + 1.36) 5 3
53 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.26 + 1.35, 1.18 + 1.45) 5 3
54 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.26 + 1.35, 1.18 + 1.36) 5 3
55 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.26 + 1.35, 1.18 + 1.27) 5 3
56 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.27 + 1.36 + 1.45) 5 3
57 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.27 + 1.45) 5 3
58 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17, 1.27 + 1.36 + 1.45) 5 3
59 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.27 + 1.45) 5 3
60 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.27 + 1.36 + 1.45) 6 3
61 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.45) 6 3
62 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.27 + 1.36) 6 3
63 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.45) 6 3
64 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26, 1.18 + 1.27 + 1.36) 6 3
65 (0, 0, 1.12, 1.13, 1.14, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.26 + 1.35, 1.18 + 1.27) 6 3
66 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17, 1.18 + 1.36 + 1.45) 6 3
67 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.45) 6 3
68 (0, 0, 1.12, 1.13, 1.14 + 1.23, 0, 1.16 + 1.25 + 1.34, 1.17 + 1.35, 1.18 + 1.45) 6 3