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Attention: using non-tested optimization, please double-check by hand. I have just 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 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^{10}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
10.

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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 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^{5}_1 is not realizable. The absolute Dynkin indices of the sl(2) subalgebras of the original summand I computed to be:
5.

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Attention: using non-tested optimization, please double-check by hand. I have just rejected type 2A^{2}_1+A^{1}_1 as non-realizable for the following reasons. I computed that the type's summand 2A^{2}_1 has complement summand A^{1}_1. 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^{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, 4.

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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 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}_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.