Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{30}_1 as non-realizable for the following reasons. The type's summand A^{30}_1 has complement summand A^{30}_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^{30}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
30.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{30}_1+A^{6}_1 as non-realizable for the following reasons. The type's summand A^{6}_1 has complement summand A^{30}_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^{30}_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.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{30}_1+A^{5}_1 as non-realizable for the following reasons. The type's summand A^{5}_1 has complement summand A^{30}_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^{30}_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.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{21}_1+A^{9}_1 as non-realizable for the following reasons. The type's summand A^{9}_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^{9}_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^{20}_1+A^{10}_1 as non-realizable for the following reasons. The type's summand A^{10}_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 A^{20}_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.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{20}_1+A^{8}_1 as non-realizable for the following reasons. The type's summand A^{8}_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 A^{20}_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.

Attention: using non-tested optimization, please double-check by hand. I have just 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 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^{9}_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^{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.

Attention: using non-tested optimization, please double-check by hand. I have just 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 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^{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^{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.

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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{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+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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{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 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}_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^{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^{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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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.

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 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^{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 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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{2}_2+A^{1}_2+A^{9}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{9}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{2}_2+A^{1}_2+A^{9}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
9, 8, 17, 2, 11.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{2}_2+A^{1}_2+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{8}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{2}_2+A^{1}_2+A^{8}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8, 16, 2, 10.

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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{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 A^{1}_4+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}_4. 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}_4+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}_4+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand A^{1}_4. 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}_4+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}_4+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}_4. 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}_4+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 D^{1}_4+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand D^{1}_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 D^{1}_4+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 D^{1}_4+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_1 has complement summand D^{1}_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 D^{1}_4+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_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^{1}_3. Then I computed the latter complement summand has centralizer 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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.

Attention: using non-tested optimization, please double-check by hand. I have just 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 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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.

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 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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^{2}_2 as non-realizable for the following reasons. I computed that the type's summand B^{2}_2 has complement summand A^{1}_3. Then I computed the latter complement summand has centralizer 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_3+B^{2}_2 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
20, 4, 2.

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 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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+2A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{8}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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+2A^{8}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8, 16.

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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{5}_1+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1+A^{3}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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, 5, 8.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_2+A^{5}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1+A^{2}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{5}_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, 5, 7.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type A^{1}_2+A^{5}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{5}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{1}_2+A^{5}_1+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5, 4, 9, 1, 6.

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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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+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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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+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 A^{1}_2+A^{4}_1+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{4}_1+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{1}_2+A^{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, 2, 6.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{1}_2+A^{28}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{28}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{1}_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, 4, 32, 1, 29.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{1}_2+A^{20}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{20}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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, 4, 24, 1, 21.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{1}_2+A^{12}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{12}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{1}_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, 4, 16, 1, 13.

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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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 2A^{1}_2+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{1}_2+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{1}_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, 4, 5, 2, 8, 9, 6, 3.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type B^{1}_2+A^{10}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand B^{1}_2+A^{10}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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 B^{1}_2+A^{10}_1+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 20, 2, 12, 1, 11.

Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2B^{1}_2+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2B^{1}_2 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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 2B^{1}_2+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 2, 1, 20, 12, 11, 4, 3.

I have rejected type 2A^{30}_1 as non-realizable for the following reasons. The type's summand A^{30}_1 has complement summand A^{30}_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^{30}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
30.

I have rejected type A^{30}_1+A^{6}_1 as non-realizable for the following reasons. The type's summand A^{6}_1 has complement summand A^{30}_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^{30}_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^{30}_1+A^{5}_1 as non-realizable for the following reasons. The type's summand A^{5}_1 has complement summand A^{30}_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^{30}_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 A^{21}_1+A^{9}_1 as non-realizable for the following reasons. The type's summand A^{9}_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^{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^{20}_1+A^{10}_1 as non-realizable for the following reasons. The type's summand A^{10}_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 A^{20}_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^{20}_1+A^{8}_1 as non-realizable for the following reasons. The type's summand A^{8}_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 A^{20}_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^{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 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^{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 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^{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 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^{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^{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^{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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{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+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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{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^{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}_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^{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^{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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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 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}_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^{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 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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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}_2+A^{1}_2+A^{9}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{9}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{2}_2+A^{1}_2+A^{9}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
9, 8, 17, 2, 11.

I have rejected type A^{2}_2+A^{1}_2+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_2+A^{8}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{2}_2+A^{1}_2+A^{8}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8, 16, 2, 10.

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 A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{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 A^{1}_4+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}_4. 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}_4+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^{1}_4+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand A^{1}_4. 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}_4+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}_4+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}_4. 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}_4+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 D^{1}_4+A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand A^{8}_1 has complement summand D^{1}_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 D^{1}_4+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 D^{1}_4+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{2}_1 has complement summand D^{1}_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 D^{1}_4+A^{2}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
2.

I have rejected type A^{1}_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^{1}_3. Then I computed the latter complement summand has centralizer 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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 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 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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 A^{1}_3+B^{2}_2 as non-realizable for the following reasons. I computed that the type's summand B^{2}_2 has complement summand A^{1}_3. Then I computed the latter complement summand has centralizer 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_3+B^{2}_2 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
20, 4, 2.

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 2A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 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}_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+2A^{8}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{8}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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+2A^{8}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
8, 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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{5}_1+A^{3}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1+A^{3}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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, 5, 8.

I have rejected type A^{1}_2+A^{5}_1+A^{2}_1 as non-realizable for the following reasons. I computed that the type's summand A^{5}_1+A^{2}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{5}_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, 5, 7.

I have rejected type A^{1}_2+A^{5}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{5}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{1}_2+A^{5}_1+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5, 4, 9, 1, 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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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+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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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+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+A^{4}_1+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{4}_1+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{1}_2+A^{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, 2, 6.

I have rejected type 2A^{1}_2+A^{28}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{28}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{1}_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, 4, 32, 1, 29.

I have rejected type 2A^{1}_2+A^{20}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{20}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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, 4, 24, 1, 21.

I have rejected type 2A^{1}_2+A^{12}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_2+A^{12}_1 has complement summand A^{1}_2. Then I computed the latter complement summand has centralizer 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{1}_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, 4, 16, 1, 13.

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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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 2A^{1}_2. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
4, 1, 8, 5, 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^{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 2A^{1}_2+2A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{1}_2+A^{1}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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^{1}_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, 4, 5, 2, 8, 9, 6, 3.

I have rejected type B^{1}_2+A^{10}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand B^{1}_2+A^{10}_1 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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 B^{1}_2+A^{10}_1+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 20, 2, 12, 1, 11.

I have rejected type 2B^{1}_2+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2B^{1}_2 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer A^{1}_5. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
35, 20, 11, 10, 8, 5, 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 2B^{1}_2+A^{1}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 2, 1, 20, 12, 11, 4, 3.