Here $V $ is finite dimensional over $\mathbb{C}$.
I am trying to show that if a Lie algebra $L \subset \mathfrak{sl}(V)$ with irreducible natural representation has an abelian ideal $I$ then $I=\{ 0 \} $. I know that for any $\phi \in I $ that $V=\ker (\phi ^{\dim V }) $ is just the generalised eigenspace for $0$ and so every $\phi \in I $ is nilpotent.
Now letting $\phi \in I $, I need to show that $ \phi (v)=$ for all $v \in V $ but I am unsure how to start this to be honest.