I'm reading Lie Groups, Lie Algebras , and Representations (first edition) by Hall and I'm stuck by a theorem (The author did not prove it):
Theorem 6.6 A complex Lie algebra is semisimple if and only if it is isomorphic to the complexification of the Lie algebra of a simply-connected compact matrix Lie group.
Through this theorem (assume it is right) we can prove that $\mathfrak{sl}(2,\mathbb{C})$ is semisimple, but $\mathfrak{sl}(2,\mathbb{C})$ is in fact simple. There is a proof : Example ideal of $\mathfrak{sl}(2,\mathbb{C})$
So whether is the theorem right or wrong? If it is wrong, is there any correct theorem related to it?
From your question, one could assume that there's some contradiction somewhere. Where is it? Yes, $\mathfrak{sl}(2,\mathbb{C})$ is simple and, yes, $\mathfrak{sl}(2,\mathbb{C})$ is semisimple. That's not a problem, since every simple Lie algebra is also semisimple.