Cohomological rigidity of solvable Lie algebras of maximal ran

We study the second cohomology group with coefficients in the adjoint module for a class of solvable Lie algebras $\mathcal{R}_{\mathcal{T}}$ that arise as maximal solvable extensions of nilpotent Lie algebras $\mathcal{N}$ of maximal rank. Under suitable structural assumptions on the root system determined by the action of a maximal torus $\mathcal{T}$ on $\mathcal{N}$, we obtain sufficient conditions for the cohomological rigidity of $\mathcal{R}_{\mathcal{T}}$. Conversely, we identify explicit configurations of roots that force the second cohomology group to be non-trivial, thereby producing broad families of solvable Lie algebras that are not cohomologically rigid. Our results extend the classical sufficient conditions of Leger and Luks, and they provide a unified and computationally effective framework for determining the cohomological rigidity of a wide class of solvable Lie algebras, including several known results.