Nichols algebras over groups with finite root system of rank two III

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We compute the finite-dimensional Nichols algebras over the sum of two simple Yetter–Drinfeld modules V and W over non-abelian epimorphic images of a certain central extension of the dihedral group of eight elements or SL(2,3), and such that the Weyl groupoid of the pair (V,W) is finite. These central extensions appear in the classification of non-elementary finite-dimensional Nichols algebras with finite Weyl groupoid of rank two. We deduce new information on the structure of primitive elements of finite-dimensional Nichols algebras over groups.