I have rejected type A^{12}_1+A^{8}_1 as non-realizable for the following reasons. The type's summand A^{8}_1 has complement summand A^{12}_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^{12}_1+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^{11}_1+A^{9}_1 as non-realizable for the following reasons. The type's summand A^{9}_1 has complement summand A^{11}_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 A^{11}_1+A^{9}_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^{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 B^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 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 B^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 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 2A^{4}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{4}_1 has complement summand A^{2}_1. Then I computed the latter complement summand has centralizer B^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 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 2A^{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:
4, 8.
I have rejected type 2A^{4}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{4}_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 2A^{4}_1+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
4, 8.
I have rejected type A^{2}_2+A^{12}_1 as non-realizable for the following reasons. I computed that the type's summand A^{12}_1 has complement summand A^{2}_2. 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^{2}_2+A^{12}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
12.
I have rejected type A^{2}_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^{2}_2. 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^{2}_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^{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}_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^{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 B^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
10, 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+A^{1}_2 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2 has complement summand A^{2}_2. 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^{2}_2+A^{1}_2 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
4, 1.
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.