Derived algebra nilpotence criterion
= 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>.