Exercise 2, § 7

Let $g$ be a Lie algebra and $n$ its largest nilpotent ideal. There exist a finite-dimensional vector space $V$ and an isomorphism of $g$ onto a subalgebra of $\mathfrak{sl}(V)$, which maps every element of $n$ to a nilpotent endomorphism of $V$. (Use Ado’s Theorem and Exercise 5 of § 1.)