Let us consider a semi-simple Lie Algebra $L$ and its Cartan subalgebra $H$, which is defined as the maximal nilpotent abelian subalgebra of $L$ such that all its elements are diagonalizable in the Adjoint representation. So, let's choose the basis where $ad_X$ are diagonals, where $X\in H$.
Now, thank's to Engel, we know that an algebra $H$ is nilpotent if its elements are ad-nilpotent, then
$$ (ad_{X})^n Y = 0, \text{for } X \in H \text{ and } Y \in H $$
I don't understand how we can have $(ad_{X})^n =0$ for $X$ in the Cartan subalgebra if these matrices are diagonal.