Nilpotent Lie algebras need not act nilpotently
= Nilpotent Lie algebras need not act nilpotently
The <Abelian Lie algebra> $kI\subseteq\operatorname{End}(V)$, for $V\ne0$, is a <Nilpotent Lie algebra> but contains the nonnilpotent <identity map>. Thus nilpotence of the abstract <Lie algebra> does not imply that a particular <Lie algebra representation> consists of <nilpotent endomorphisms>. <Engel theorem> requires nilpotence of every represented endomorphism to obtain strict upper triangularity.