There is also section 7.5 and Table VI in the textbook of J. Fucs and C. Schweigert where quadratic form matrices G (G is inverse of symmetrized Cartan matrix) are presented. For simply laced (ADE) case $G = A^{-1}$.

In the $D_4$-case (with the group of symmetry of Dynkin diagram $S_3$) the matrix $2 A^{-1}$ is integer valued.

For example, for $A_3$ \begin{equation} \label{1} P = \left( \begin{array}{cccccc} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{array} \right). \end{equation} corresponds to the permutation of nodes $(1,2,3) \mapsto (3,2,1)$ (which is a generator of $Z_2$-group of symmetry of Dynkin diagram).

The question is motivated by certain physical problem, where some integer valued matrix $B$ appears. For the case (a): $B = 2A^{-1}$ and for the case (b): $B = A^{-1} (I + P)$. As we expected $(a)$ is a well-known fact. The conjecture (b) was verified for $E_6$. For classical series $A_r$, $D_r$ the (b) was verified by MATHEMATICA for some ranks $r$ ($r > 4$ for $D$-series).

Dear Robert Israel, thank you very much! Your proof is done in a geometric style which is close (to my opinion) to that of V.I. Arnol'd.

In Bourbaki definition of manifold the following order of notions is used: set, manifold, topological space. This is contrary to the definition of manifold in some other textbooks, where another order of notions is used: set, topological space, manifold.

To my memory I had a simple proof (in summer or spring) that this is the final list. (The proof is of the school olympiade level). I will present it when I will have a time and/or I will prepare the paper which will use this example.

Dear GA316, I have deleted my remark. I apologize a lot.

Many thanks! I apologize that I asked the question without any preliminary studying of the subject (I was reading in physical papers about possible presence of continuous spectrum in some cases but have doubts about generality of that). It looks that the same situation may take place for general case, when $B$ is non-compact but of finite volume and arbitrary $n$. It will be good to have also a standard citation about the classical fact you have mentioned for the compact case.