 Two-dimensional subalgebra criterion for Lie algebra nilpotence

Over an algebraically closed field, a finite-dimensional Lie algebra is nilpotent exactly when every two-dimensional Lie subalgebra is abelian. A nonnilpotent Adjoint representation matrix has a nonzero eigenvalue $\lambda$ with eigenvector $y$, so $[x,y]=\lambda y$ generates a nonabelian two-dimensional subalgebra. Conversely a subalgebra of a nilpotent Lie algebra is nilpotent, while a nonabelian two-dimensional algebra has a basis $u,v$ with $[u,v]=v$ and a nonnilpotent adjoint matrix. Algebraic closure is what guarantees the required eigenvector.