The question is in the title: how can I show $\mathfrak{sl}_n(\mathbb{C})$ is simple?
In every book I scoured, they say $\mathfrak{sl}_n(\mathbb{C})$ is simple but they do not provide a proof! Is there a proof which does not resort to Lie groups?
Erdmann and Wildon book on Lie algebras, this is an exercise and it appears quite early on, before proving theorems like Cartan's Cariterion or before developing Killing forms. So, is there a way to show the result in a fairly basic way?
Thanks.