Engel's theorem
Suppose every element of a Lie algebra, when it acts on the algebra by bracketing, is a nilpotent operator — apply it enough times and you get zero. Intuitively each generator is individually weak. Engel's theorem says this individual weakness conspires into collective weakness: the whole algebra must be nilpotent. Local nilpotence forces global nilpotence.
Stated for a finite-dimensional Lie algebra L: L is nilpotent if and only if ad(x) is a nilpotent endomorphism for every x in L. The hard direction is the if part. The key linear-algebra lemma behind it: if L is a Lie algebra of nilpotent operators on a nonzero finite-dimensional vector space V, then there is a nonzero vector v killed by all of them, Lv = 0; iterating produces a flag making every operator strictly upper-triangular.
The condition is only about ad-nilpotence, an intrinsic property, yet it yields a representation-theoretic conclusion (common eigenvector, simultaneous strict triangularization). Engel's theorem is the nilpotent-case anchor; Lie's theorem is its solvable-case companion, weakening the hypothesis but also the conclusion and adding an algebraically-closed field of characteristic zero.
Let L be the strictly upper-triangular n-by-n matrices. Each matrix is nilpotent, and ad(x)(y) = xy - yx is nilpotent too. Engel's theorem then guarantees L is a nilpotent Lie algebra — consistent with L^{(n)} terms eventually reaching zero.
Strictly upper-triangular matrices: ad-nilpotent everywhere, hence nilpotent by Engel.
Watch the difference: every ad(x) being nilpotent is required, not just nilpotency of some elements. An element x with ad(x) nilpotent is called ad-nilpotent; Engel asks all of them to be.