Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{10}_1 as non-realizable for the following reasons. The type's summand A^{10}_1 has complement summand A^{10}_1. I computed the latter complement summand has centralizer A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
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 2A^{10}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{5}_1 as non-realizable for the following reasons. The type's summand A^{5}_1 has complement summand A^{5}_1. 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 2A^{5}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{5}_1+A^{3}_1 as non-realizable for the following reasons. The type's summand A^{3}_1 has complement summand A^{5}_1. 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 A^{5}_1+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^{3}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{3}_1 has complement summand A^{1}_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^{3}_1+A^{1}_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}_1+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_1+A^{1}_1 has complement summand A^{1}_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}_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, 2, 3.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_3+A^{10}_1 as non-realizable for the following reasons. I computed that the type's summand A^{10}_1 has complement summand A^{1}_3. Then I computed the latter complement summand has centralizer A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
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^{10}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_3+B^{1}_2 as non-realizable for the following reasons. I computed that the type's summand B^{1}_2 has complement summand A^{1}_3. Then I computed the latter complement summand has centralizer A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
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+B^{1}_2 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 2, 1.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_2+A^{4}_1+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{4}_1+A^{3}_1 has complement summand A^{1}_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^{1}_2+A^{4}_1+A^{3}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
3, 4, 7.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_2+A^{4}_1+A^{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^{1}_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^{1}_2+A^{4}_1+A^{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 2A^{1}_2+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{3}_1 has complement summand A^{1}_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 2A^{1}_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, 4, 7, 1.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{1}_2+A^{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^{1}_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 2A^{1}_2+A^{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.
Attention: using non-tested optimization, please double-check by hand. I have just rejected type 4A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 3A^{1}_1 has complement summand A^{1}_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^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
1, 2, 3.