Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{9}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{9}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{9}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
9.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{9}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{9}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{9}_1+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
9.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{8}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{8}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{8}_1+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{8}_1+2A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
1, 8, 9.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{6}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{6}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{6}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
6.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{4}_1 as non-realizable for the following reasons. The type's summand A^{4}_1 has complement summand A^{4}_1. I computed the latter complement summand has centralizer A^{2}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
8, 2. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type 2A^{4}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
4.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{4}_1+2A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{4}_1+A^{2}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{4}_1+2A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2, 4, 6.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{2}_2+A^{28}_1 as non-realizable for the following reasons. I computed that the type's summand A^{28}_1 has complement summand A^{2}_2. Then I computed the latter complement summand has centralizer A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+A^{28}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
28.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{2}_2+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{3}_1 has complement summand A^{2}_2. Then I computed the latter complement summand has centralizer A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+A^{3}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
3.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{2}_2+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8, 2.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{2}_2+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+2A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
1, 8, 9, 2, 3.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type 4A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand 3A^{2}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type 4A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2, 4, 6.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_3+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_3 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_3+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 4, 2, 1.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type B^{1}_3+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand B^{1}_3. Then I computed the latter complement summand has centralizer 0. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type B^{1}_3+A^{8}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_2+A^{6}_1 as non-realizable for the following reasons. I computed that the type's summand A^{6}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer A^{2}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
8, 2. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_2+A^{6}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
6.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_2+A^{4}_1 as non-realizable for the following reasons. I computed that the type's summand A^{4}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer A^{2}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
8, 2. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_2+A^{4}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
4.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_2+2A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{2}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_2+2A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2, 4, 6, 1, 3.
I have rejected type A^{9}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{9}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{9}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
9.
I have rejected type A^{9}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{9}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{9}_1+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
9.
I have rejected type A^{8}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{8}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8.
I have rejected type A^{8}_1+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{8}_1+2A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
1, 8, 9.
I have rejected type A^{6}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{6}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{6}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
6.
I have rejected type 2A^{4}_1 as non-realizable for the following reasons. The type's summand A^{4}_1 has complement summand A^{4}_1. I computed the latter complement summand has centralizer A^{2}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
8, 2. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type 2A^{4}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
4.
I have rejected type A^{4}_1+2A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{4}_1+A^{2}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{4}_1+2A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2, 4, 6.
I have rejected type A^{2}_2+A^{28}_1 as non-realizable for the following reasons. I computed that the type's summand A^{28}_1 has complement summand A^{2}_2. Then I computed the latter complement summand has centralizer A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+A^{28}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
28.
I have rejected type A^{2}_2+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{3}_1 has complement summand A^{2}_2. Then I computed the latter complement summand has centralizer A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+A^{3}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
3.
I have rejected type A^{2}_2+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8, 2.
I have rejected type A^{2}_2+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{2}_2+2A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
1, 8, 9, 2, 3.
I have rejected type 4A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand 3A^{2}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type 4A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2, 4, 6.
I have rejected type A^{1}_3+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_3 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 11, 10, 8, 3, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_3+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 4, 2, 1.
I have rejected type B^{1}_3+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand B^{1}_3. Then I computed the latter complement summand has centralizer 0. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type B^{1}_3+A^{8}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8.
I have rejected type A^{1}_2+A^{6}_1 as non-realizable for the following reasons. I computed that the type's summand A^{6}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer A^{2}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
8, 2. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_2+A^{6}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
6.
I have rejected type A^{1}_2+A^{4}_1 as non-realizable for the following reasons. I computed that the type's summand A^{4}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer A^{2}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
8, 2. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_2+A^{4}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
4.
I have rejected type A^{1}_2+2A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{2}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer A^{1}_3. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 4, 2, 1. If the type was realizable, those would have to contain the absolute Dynkin indices of sl(2) subalgebras of the original summand. However, that is not the case. I can therefore conclude that the Dynkin type A^{1}_2+2A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2, 4, 6, 1, 3.