  We compute the finite-dimensional Nichols algebras over the sum of two simple
Yetter-Drinfeld modules V and W over non-abelian quotients 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.
