We defined a Lie algebra to be simple, if it has no proper Lie ideals and is not $k$ (the ground field). We have the proposition that $\mathfrak{sl}_n(\mathbb{R})$ and $\mathfrak{sl}_n(\mathbb{C})$ are simple $\mathbb{R}$-Lie algebras, and $\mathfrak{sl}_n(\mathbb{C})$ is also simple as a $\mathbb{C}$-Lie algebra. We have shown the proposition for the case $n=2$ using an $\mathfrak{sl}_2$-triple.
How would I show the statement for general $n \in \mathbb{N}$?
I suspect there's a way to use induction, but don't see how that would work. Also, I have no idea how I would generalize and apply the idea with the $\mathfrak{sl}_2$-triple to higher dimensions. I can't even just do the case $n=3$. How do I generalize?
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$\begingroup$There is a direct proof that $\mathfrak{sl}(n,K)$ is a simple Lie algebra for any field $K$ of characteristic zero, which just uses Lie brackets of traceless matrices to show that a nontrivial ideal $J$ must be $\mathfrak{sl}(n,K)$ itself, see 6.4 in the book "Naive Lie Theory". This works uniformly for all $n\ge 2$. For a proof using a bit more theory, see Lemma $1.3$ here.
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