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