Derived algebra nilpotence criterion

A finite-dimensional complex Lie algebra is a Solvable Lie algebra exactly when its derived algebra is a Nilpotent Lie algebra. The forward direction follows from the Lie theorem in the Adjoint representation and lifting nilpotence through its central kernel. The reverse direction follows because the derived series of a Lie algebra after its first term is the derived series of a Lie algebra of the derived algebra.