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

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I have rejected type A^{1}_3+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_3 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer B^{1}_2+A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 10, 11, 2, 3. 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}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 4, 2, 1.

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I have rejected type A^{1}_3+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand A^{1}_3 has complement summand A^{1}_1. Then I computed the latter complement summand has centralizer B^{1}_2+A^{1}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
1, 10, 11, 2, 3. 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}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10, 4, 2, 1.

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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^{2}_1. Then I computed the absolute Dynkin indices of the centralizer's sl(2)-subalgebras, namely:
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.