Suppose $\mathfrak{g}$ be a Lie algebra over $\mathbb{F}$. Then $\mathfrak{g}$ is
nilpotent if and only if, for all $x \in \mathfrak{g}$, $\mathrm{ad}~ x$ is a nilpotent linear operator on $\mathfrak{g}$.
This is Engel's theorem
My doubt is this:
Suppose $\mathfrak{g}$ be a Lie algebra consisting of nilpotent operators on a finite vector space $V$. Can we say that $\mathfrak{g}$ is nilpotent ?
Best Answer
Let $\mathfrak{g}$ be a Lie algebra consisting of nilpotent operators on a finite vector space $V$. Then $\mathfrak{g}$ is a subalgebra of $\mathfrak{gl}(V)$. Since $x$ nilpotent implies that $\operatorname{ad}(x)$ is nilpotent, we have that $\operatorname{ad}(x)$ is nilpotent for all $x\in \mathfrak{g}$. By Engel's Theorem $\mathfrak{g}$ is nilpotent.