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In the present paper we study SKT and generalized K\"ahler structures on solvable Lie algebras with (not necessarily abelian) codimension two nilradical. We treat separately the case of J-invariant nilradical and non J-invariant nilradical. A classification of such SKT Lie algebras in dimension six is provided. In particular, we give a general construction to extend SKT nilpotent Lie algebras to SKT solvable Lie algebras of higher dimension, and we construct new examples of SKT and generalized K\"ahler compact solvmanifolds.
Brienza et al. (Tue,) studied this question.
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