Sum of nilpotent ideals in a Lie algebra is nilpotent
We have $(I+J)^{2m}\subseteq I^m+J^m$ for all $m\ge 1$, hence $(I+J)^{2m}=0$ for $m$ large enough. I find this proof very natural, but indeed there is another proof using Engel's theorem and another two lemmas from representation theory; so it is perhaps more elegant, but it is also more complicated. This depends on your taste, too. For the alternative proof see Lemma $5.3.3$, Lemma $5.3.4$ and Lemma $5.3.5$ here.