The goal is to prove directly that $\mathfrak{sl}(n,\Bbb C)$ is a simple Lie algebra for $n \ge 2$
So far I'm trying to think of ways to show that there are no non-trivial ideals (definition of simple) for $\mathfrak{sl}(n,\Bbb C)$, where $\mathfrak{sl}(n,\Bbb C)$ is the collection of all matrices which have trace $= 0$.
I'm still trying to wrap my head around Lie algebras, so any help is appreciated.