Derived algebra ($\mathcal{D}(\mathfrak{g})=[\mathfrak{g},\mathfrak{g}]$)

The derived algebra is the linear span of all Lie brackets $[x,y]$ in a Lie algebra. The Jacobi identity makes it an ideal of a Lie algebra. Its quotient is the largest Abelian Lie algebra quotient of $\mathfrak{g}$.

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.