Are subalgebras of simple Lie algebras simple?

Question

Do subalgebras of simple Lie algebras have to be simple too?

Answer 1

Not at all. Actually, every finite-dimensional Lie algebra $\mathfrak g$ over a field $F$ is isomorphic to a subalgebra of some $\mathfrak{sl}(n,F)$. That's because, by Ado's theorem, $\mathfrak g$ is isomorphic to a subalgebra of $\mathfrak{gl}(n,F)$, which, in turn, is isomorphic to a subalgebra of $\mathfrak{sl}(n+1,F)$.