The formulation is somewhat out of focus, starting with the notation
$a_n(x)$ for characteristic polynomial (what is
$n$?). The roots indicated do occur in Coxeter's formulation, but not as the eigenvalues of your matrix
$A$. It would help in any case to quote your own source and to describe the simplest nontrivial example involving a
$2 \times 2$ matrix.
The original source is probably the influential paper by Coxeter in Duke Math. J.
18 (1951), which doesn't actually deal with simple complex Lie algebras and their Cartan matrices. Instead the framework is the study of finite real reflection groups (including Weyl groups of simple Lie algebras as a special case). So the results on Coxeter elements and exponents apply more broadly to reflection groups which need not be crystallographic. I'd have to look more carefully at the paper, but my impression is that what you are looking for doesn't require case-by-case study. There is a version of this development in the lengthy Exercise 3 (applied in Exercise 4) for Chapter V, Section 6, in Bourbaki Groupes et algebres de Lie (1968). Note that this chapter in Bourbaki just deals with reflection groups, before a treatment of crystallographic root systems and the related classification of finite Coxeter groups in Chapter VI.
P.S. While the case-by-case calculation of exponents for a finite Coxeter group (and their relationship with degrees of fundamental invariants) has evolved since Coxeter's original work, the matrix manipulations involved in the question here don't require knowing the explicit values of the
$m_i$ in each case. Anyway, the question really has nothing directly to do with simple Lie algebras but only with finite Coxeter groups and Coxeter elements.