Are the Lie algebras $\mathfrak{sl}_2(\Bbb{R})$ and $\mathfrak{su}(2)$ simple?

Question

I believe I have proven that the real Lie algebras $\mathfrak{sl}_2(\Bbb{R})$ and $\mathfrak{su}(2)$ are simple. However I have tried searching for results that mention this but they only talk of these Lie algebras as being semi-simple. Is what I have proven false, or is it simply that these Lie algebras are simple and hence semi-simple?

My proof of simplicity uses the fact that we have explicit bases for these two lie algebras. I then proceed to show that there are no $1$ - dimensional ideals, and then using the fact that the Killing form is ad-invariant I know that any $2$ - dimensional ideal would give rise to a one dimensional ideal, namely its orthogonal complement. This contradicts the fact that there are no 1-dimensional ideals, and so the real Lie algebras above are simple.

Answer 1

Yes they are simple. Semi-simple complex Lie algebras are classified by their Dynkin diagrams and the connected Dynkin diagrams correspond to simple Lie algebras. The complexifications of sl(2) and su(2) is sl(2,C) which has a connected Dynkin diagram and is therefore simple. It follows that its real forms are also simple.