Abstract We use the shear construction to construct and classify a wide range of two-step solvable Lie groups admitting left-invariant SKT structure. reduce this specification data on Abelian algebras, which then is studied more deeply in different cases. obtain classifications structure results for $$\mathfrak {g}$$ g <...

Let G be a group, and $${{\,\mathrm{Sol}\,}}(G)=\{x \in : \langle x,y \rangle \text { is solvable for all } y G\}$$ . We associate graph $$\mathcal {NS}_G$$ (called the non-solvable of G) with whose vertex set $$G \setminus {{\,\mathrm{Sol}\,}}(G)$$ two distinct vertices are adjacent if they generate subgroup. In this paper, we study many properties particular, obtain results on degree, cardina...