= 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.
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