Strong Kähler with torsion solvable lie algebras with codimension 2 nilradical

In this paper, we study strong K & auml;hler with torsion (SKT) and generalized K & auml;hler structures on solvable Lie algebras with (not necessarily abelian) codimension 2 nilradical. We treat separately the case of J$J$-invariant nilradical and non-J$J$-invariant nilradical. A classification of such SKT Lie algebras in dimension 6 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 & auml;hler compact solvmanifolds.