Root systems and orthogonal groups of root lattices (Q795911)

For an indecomposable symmetrizable generalized Cartan matrix of spherical, Euclidean or hyperbolic (more appropriately, minimal hyperbolic) type with corresponding (generalized) root system \(\Delta\), the lattice \(\Gamma\) generated by \(\Delta\) is equipped with a canonical quadratic form. In the spherical case only for type \(C_4\) is the orthogonal group \(O(\Gamma)\) not generated by the Weyl group, the diagram automorphisms and the multiplication by \(-1\), but for the other cases this happens more often. As main theorem it is proved that the union of the images of \(\Delta\) under the elements of \(O(\Gamma)\) is again a root system; for several types the Cartan matrix of this root system is determined.