I found a observation in the beginning of the proof of Engel's Theorem in the Fulton's book "Representation Theory"
Observation: if $X \in \mathfrak{gl}(V)$ is any nilpotent element, then the action $\operatorname{ad}(X):\mathfrak{gl}(V) \to \mathfrak{gl}(V)$ is nilpotent. This is straightforward: to say that $X$ is nilpotent is to say that there exist a flag of a subspaces: $0 \subset V_1 \subset V_2 ... \subset V_{k+1}=V$ such that $X(V_i) \subset V_{i-1}$; we can then check that for any endomorphism $Y$ of $V$ the endomorphism $\operatorname{ad}(X)^m (Y)$ carries $V_i$ into $V_{i+k-m}$.
Is this index $i+k-m$ correct? I tried to check the last sentence, but I can't.