Trivially, for any Lie Algebra (LA) g, g':=[g,g] is an ideal. What's wrong with the following argument?
Be g a simple LA, then it has to be g'=g by definition of simple LA. But [g,g]=g seems to be an alternative way of characterizing a semi simple LG. Furthermore, for sl(2) it doesn't seem to be true z=[x,y] for any x,y,z € g. Thus, the previous implication I'm inclined to make must be flawed, but I can't see my mistake.
I'm following J. Fuchs book. Thanks in advance.
A Lie algebra $L$ satisfying $[L,L]=L$ is called perfect. In particular, semisimple Lie algebras are perfect. The converse, however is not true in general. It is easy to see that the semidirect product of $\mathfrak{sl}_n$ with an irreducible representation $V$ of it becomes a perfect Lie algebra, which is not semisimple. So $[L,L]=L$ is definitely not an alternative way to characterize semisimple Lie algebras.
@Tobias Sorry, I expected an email notification for any comment/answer I might get here, but that wasn't the case and I missed your comment.
Meanwhile, I sorted out my original misunderstanding that gave rise to this question. Being a semi simple LA means that every element z can be written as the commutator of two other elements, say, x,y, i.e., z=[x,y]. This, however, is a more stringent condition than saying [g,g]=g, at least in general. I was carried over by the notation and didn't consider that [g,g] is defined as the span of commutators.
Thus, for any simple LA it is [g,g]=g as it has no other ideals besides 0 and g. Furthermore, this is also true for semi simple LAs.
I have not seen stated anywhere, in particular, not within the first five chapters of J. Fuchs' book, that sl(2) can also be seen as a semi simple LA in the sense that $\forall z \in sl(2), z=[x,y]\,;\,x,y \in sl(2)$. sl(2) it's just the smallest simple LA there is -if I got that right.
In any case, thanks for your quick reply.
