In chapter 1 of Kac's book "Infinite dimensional Lie algebras" it is mentioned that two Kac-Moody algebras are isomorphic if and only if their Cartan matrices are isomorphic (i.e. they are the same up to permutation). A reference to the article "Infinite flag varieties and conjugacy theorems" by Peterson and Kac is given, where it is proven that Cartan subalgebras are conjugate (which implies this result). But in this article it is assumed that the Cartan matrix is symmetrizable! Now, is this result true for any Cartan matrix or just for symmetrizable ones?
At the end of the article it is mentioned that all results which do not use the bilinear form also hold for a general Cartan matrix but I don't want to go through this whole article.
