Thank you very much again, this self-contained argument saves the day for me (I know very little of the general theory of Lie algebras). It was very helpful your answer.

Everything makes sense, the only I would ask is a little explanation as to the fact that $\mathfrak{a}+\mathfrak{h}=\mathfrak{g}$, is it always true that a cartan subalgebra plus the derived subalgebra fill the total algebra?

okey that was silly, any 2-dimensional nilpotent algebra is abelian

Thank you very much for the answer. A couple of questions: the fact that $\mathfrak{h}$ is 2-dimensional follows since in $\mathfrak{g}$ there are not nilpotent subalgebras of dimension $\ge 3$ (at the complex level), right? Another: why do you get an abelian subalgebra of $\mathfrak{gl}_2(K)$? The algebra $\mathfrak{h}$ is nilpotent, in principle it is not abelian, right?