  By using the concept of weight graph associated to certain nilpotent Lie
algebras $\frak{g}$, we find necessary and sufficient conditions for a
semidirect product $\frak{g}\oplus T_{i}$, where $T_{i}<T$ is a subalgebra of a
maximal torus of derivations $T$ of $\frak{g}$ which induces a decomposition of
$\frak{g}$ into one dimensional weight spaces, to be 2-step solvable. In
particular we show that the semidirect product of such a Lie algebra with its
torus of derivations cannot be itself 2-step solvable.
