Let $\mathfrak{g}$ be a Lie Algebra (finite dimensional, over $\mathbb{C}$). Engel's theorem tells us that if there exists a $m\in \mathbb{N}$ such that $ad(x)^m = 0$, $\forall x\in \mathfrak{g}$, then $\mathfrak{g}$ is nilpotent. And if $\mathfrak{g}$ is $k$-step nilpotent (i.e. the $k$-th term of the lower central series of $\mathfrak{g}$ is the first one that is 0, or equivalently $ad(x_1)ad(x_2) \ldots ad(x_k) = 0$ $\forall x_1, \ldots, x_k \in \mathfrak{g}$), it is clear that $$\min\(m\in \mathbb{N} : ad(x)^m = 0 \forall x\in \mathfrak{g} ) \leq k.$$ Can we find an example where the previous inequality is not an equality?

If this is a very basic fact in the theory, I apologize.

`$ad(x)$`

is nilpotent. (On the other hand, combining Engel's Theorem with Ado's Theorem allows one to regard an abstract nilpotent Lie algebra as a subalgebra of the upper triangular matrices.) It might be helpful to recall more explicitly what is meant by`$k$`

-step nilpotent. – Jim Humphreys May 19 '11 at 21:18