Equivalent characterizations of nilpotency for Lie algebras

Theorem (TFAE: nilpotency)

Let $\mathfrak g$ be a finite-dimensional Lie algebra over a field of characteristic $0$. The following are equivalent.

1. Lower central series terminates: the lower central series $\mathfrak g_1=\mathfrak g$, $\mathfrak g_{k+1}=[\mathfrak g,\mathfrak g_k]$ satisfies $\mathfrak g_N=0$ for some $N$. Equivalently, $\mathfrak g$ is nilpotent .

2. Engel condition on adjoints: for every $X\in\mathfrak g$, the endomorphism $\mathrm{ad}_X:\mathfrak g\to\mathfrak g$ in the adjoint representation is nilpotent.

3. Strict upper-triangular realization: there exists an injective Lie algebra homomorphism

   $$\mathfrak g \hookrightarrow \mathfrak{gl}(N,\mathbb F)$$

   whose image consists of strictly upper triangular matrices (hence all images are nilpotent endomorphisms). This can be viewed as a refinement of Ado’s theorem specialized to the nilpotent case.

4. Central series by ideals: there exists a chain of ideals

   $$\mathfrak g=\mathfrak g^{(0)}\supset \mathfrak g^{(1)}\supset \cdots \supset \mathfrak g^{(N)}=0$$

   such that $[\mathfrak g,\mathfrak g^{(i)}]\subseteq \mathfrak g^{(i+1)}$ for all $i$.

Context

Condition (2) is the Lie-algebraic form of “all infinitesimal conjugations are nilpotent,” while (3) connects nilpotent Lie algebras to concrete matrix models such as strictly upper-triangular examples . Nilpotency is stronger than solvability; in fact nilpotent implies solvable .