I have rejected type A^{35}_1+A^{21}_1 as non-realizable for the following reasons. The type's summand A^{21}_1 has complement summand A^{35}_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^{35}_1+A^{21}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
21.

I have rejected type A^{21}_1+A^{14}_1 as non-realizable for the following reasons. The type's summand A^{14}_1 has complement summand A^{21}_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^{21}_1+A^{14}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
14.

I have rejected type A^{14}_1+A^{6}_1 as non-realizable for the following reasons. The type's summand A^{6}_1 has complement summand A^{14}_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^{14}_1+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^{11}_1+A^{10}_1 as non-realizable for the following reasons. The type's summand A^{10}_1 has complement summand A^{11}_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^{11}_1+A^{10}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10.

I have rejected type A^{11}_1+A^{3}_1 as non-realizable for the following reasons. The type's summand A^{3}_1 has complement summand A^{11}_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^{11}_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 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}_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^{10}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10.

I have 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 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^{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^{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 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^{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.

I have rejected type A^{3}_1+A^{2}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{3}_1+A^{2}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
20, 10, 5, 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^{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:
2, 3, 5.

I have 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}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
20, 10, 5, 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.

I have rejected type A^{2}_3+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_3 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
20, 10, 5, 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}_3+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
20, 8, 4, 2.

I have rejected type 2A^{2}_1+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{2}_1+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
20, 10, 5, 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 2A^{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, 4, 5.

I have 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}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
20, 10, 5, 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.

I have rejected type A^{1}_5+A^{21}_1 as non-realizable for the following reasons. I computed that the type's summand A^{21}_1 has complement summand A^{1}_5. 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^{1}_5+A^{21}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
21.

I have rejected type A^{1}_3+A^{11}_1 as non-realizable for the following reasons. I computed that the type's summand A^{11}_1 has complement summand A^{1}_3. 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}_3+A^{11}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
11.

I have 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}_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}_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.

I have rejected type 2A^{1}_3 as non-realizable for the following reasons. I computed that the type's summand A^{1}_3 has complement summand A^{1}_3. 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}_3 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 A^{1}_3+A^{1}_2+A^{21}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_3+A^{21}_1 has complement summand A^{1}_2. 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}_3+A^{1}_2+A^{21}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
21, 10, 31, 4, 25, 2, 23, 1, 22.

I have rejected type A^{1}_3+A^{1}_2+A^{6}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_3+A^{6}_1 has complement summand A^{1}_2. 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}_3+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, 10, 16, 4, 2, 8, 1, 7.

I have 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}_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}_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.

I have rejected type A^{1}_2+A^{10}_1+A^{6}_1 as non-realizable for the following reasons. I computed that the type's summand A^{10}_1+A^{6}_1 has complement summand A^{1}_2. 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+A^{10}_1+A^{6}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
6, 10, 16.

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^{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+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+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}_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+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.

I have 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}_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+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.

I have rejected type A^{1}_2+3A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand 3A^{2}_1 has complement summand A^{1}_2. 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+3A^{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 2A^{1}_2+A^{6}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{6}_1 has complement summand A^{1}_2. 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 2A^{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, 4, 10, 1, 7.

I have 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}_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 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.

I have 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}_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 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.

I have rejected type B^{1}_2+A^{1}_2+A^{21}_1 as non-realizable for the following reasons. I computed that the type's summand B^{1}_2+A^{21}_1 has complement summand A^{1}_2. 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 B^{1}_2+A^{1}_2+A^{21}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
21, 10, 31, 2, 23, 1, 22.

I have rejected type B^{1}_2+A^{1}_2+A^{6}_1 as non-realizable for the following reasons. I computed that the type's summand B^{1}_2+A^{6}_1 has complement summand A^{1}_2. 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 B^{1}_2+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, 10, 16, 2, 8, 1, 7.

I have 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}_4. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
20, 10, 5, 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.