I have rejected type A^{45}_1+A^{40}_1 as non-realizable for the following reasons. The type's summand A^{40}_1 has complement summand A^{45}_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^{45}_1+A^{40}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
40.
I have rejected type A^{37}_1+A^{8}_1 as non-realizable for the following reasons. The type's summand A^{8}_1 has complement summand A^{37}_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^{37}_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^{37}_1+A^{3}_1 as non-realizable for the following reasons. The type's summand A^{3}_1 has complement summand A^{37}_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^{37}_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.
I have rejected type A^{36}_1+A^{9}_1 as non-realizable for the following reasons. The type's summand A^{9}_1 has complement summand A^{36}_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^{36}_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^{36}_1+A^{4}_1 as non-realizable for the following reasons. The type's summand A^{4}_1 has complement summand A^{36}_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^{36}_1+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^{35}_1+A^{5}_1 as non-realizable for the following reasons. The type's summand A^{5}_1 has complement summand A^{35}_1. 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^{35}_1+A^{5}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5.
I have rejected type 2A^{20}_1 as non-realizable for the following reasons. The type's summand A^{20}_1 has complement summand A^{20}_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^{20}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
20.
I have rejected type A^{18}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{18}_1 has complement summand A^{2}_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^{18}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
18.
I have rejected type A^{12}_1+A^{9}_1 as non-realizable for the following reasons. The type's summand A^{9}_1 has complement summand A^{12}_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^{12}_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^{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 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^{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 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^{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^{10}_1+A^{8}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{10}_1+A^{8}_1 has complement summand A^{2}_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^{10}_1+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, 10, 18.
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 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^{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^{8}_1+A^{2}_1+A^{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^{2}_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+A^{2}_1+A^{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^{5}_1+A^{4}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1+A^{4}_1 has complement summand A^{2}_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^{5}_1+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:
4, 5, 9.
I have rejected type A^{5}_1+A^{4}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1+A^{4}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
84, 36, 35, 20, 12, 11, 10, 9, 8, 4, 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^{5}_1+A^{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, 5, 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 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^{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+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{4}_1+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer C^{1}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
84, 36, 35, 20, 12, 11, 10, 9, 8, 4, 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+2A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
1, 4, 5, 8, 9.
I have rejected type A^{2}_4+A^{45}_1 as non-realizable for the following reasons. I computed that the type's summand A^{45}_1 has complement summand A^{2}_4. 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 A^{2}_4+A^{45}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
45.
I have rejected type A^{2}_4+A^{5}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1 has complement summand A^{2}_4. 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 A^{2}_4+A^{5}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5.
I have rejected type A^{2}_3+A^{20}_1 as non-realizable for the following reasons. I computed that the type's summand A^{20}_1 has complement summand A^{2}_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^{2}_3+A^{20}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
20.
I have rejected type C^{1}_3+A^{5}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1 has complement summand C^{1}_3. 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 C^{1}_3+A^{5}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5.
I have rejected type A^{2}_2+A^{37}_1 as non-realizable for the following reasons. I computed that the type's summand A^{37}_1 has complement summand A^{2}_2. 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^{37}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
37.
I have rejected type A^{2}_2+A^{13}_1 as non-realizable for the following reasons. I computed that the type's summand A^{13}_1 has complement summand A^{2}_2. 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^{13}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
13.
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 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^{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^{10}_1+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{10}_1+A^{3}_1 has complement summand A^{2}_2. 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^{10}_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, 10, 13.
I have rejected type A^{2}_2+A^{10}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{10}_1 has complement summand A^{2}_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+A^{10}_1+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 8, 18, 2, 12.
I have rejected type A^{2}_2+A^{5}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1 has complement summand A^{2}_2. 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^{5}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5.
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 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^{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 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^{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+2A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{2}_1 has complement summand A^{2}_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^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2, 8, 10, 4.
I have rejected type A^{2}_2+A^{2}_1+A^{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^{2}_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+A^{2}_1+A^{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 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 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^{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+B^{1}_2+A^{18}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{18}_1 has complement summand B^{1}_2. 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+B^{1}_2+A^{18}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
18, 8, 26, 2, 20.
I have rejected type A^{2}_2+B^{1}_2+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{3}_1 has complement summand B^{1}_2. 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+B^{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, 8, 11, 2, 5.
I have rejected type A^{2}_2+B^{1}_2+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+B^{1}_2 has complement summand A^{2}_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+B^{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:
10, 2, 1, 8, 18, 9, 12, 4, 3.
I have rejected type A^{2}_2+3A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 3A^{1}_1 has complement summand A^{2}_2. 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+3A^{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.
I have rejected type B^{1}_2+A^{8}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand B^{1}_2+A^{8}_1 has complement summand A^{2}_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 B^{1}_2+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, 10, 18, 2, 1, 9.
I have rejected type 2B^{1}_2+A^{20}_1 as non-realizable for the following reasons. I computed that the type's summand B^{1}_2+A^{20}_1 has complement summand B^{1}_2. 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 2B^{1}_2+A^{20}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
20, 10, 30, 2, 22, 1, 21.