A Levi-type decomposition on two-step solvable Lie algebras with a complex structure

We prove that a large class of $2$-step solvable Lie algebras equipped with a complex structure $J$ admits a Levi-Malcev type decomposition, adapted to $J$. As an application, we prove that the Fino–Vezzoni conjecture holds true for $2$-step solvable unimodular Lie algebras. Finally, we give a structural characterisation of $2$-step, unimodular, completely solvable Lie algebras admitting an SKT metric.