 Nilpotent Lie algebras need not act nilpotently

The abelian Lie algebra $kI\subseteq \mathrm{End}(V)$, for $V\ne 0$, 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.