While reading the book James E Humphreys: "Introduction to Lie Algebras and Representation Theory". I encountered the next proposition
I understand the proof, and I was wondering if the reverse argument is true (If $\Phi$ is an irreducible Root system, then $L$ is simple). I have been thinking on it, and I don't see any problem to say the contrary, but I have been unable to prove it.
Any help would be appreciated.