I have went through a couple of books, none of which mention the validity of this. In my class we just proved the converse and we were asked to think about this.
For the past couple of weeks we have been talking about representations of sl(n,F) and just now started talking about Killing form. I don't see how to apply what we've done thus far to find a counter example, as I tried the few lie algebras I am familiar with. So I am thinking it might be true.
I do know L is semi-simple if and only if it is a direct sum of simple ideals.