Coxeter element and particle masses

Let $mathfrak{g}$ be a simple Lie algebra of rank $r$ over $mathbb{C}$, $mathfrak{h} subset mathfrak{g}$ a Cartan subalgebra. We construct a family of $r$ commuting Hermitian operators acting on $mathfrak{h}$ whose eigenvalues are equal to the coordinates of the eigenvectors of the Cartan matrix of $mathfrak{g}$.