  We explore some interesting features of the characteristic polynomial of the
Cartan matrix of a simple Lie algebra. The characteristic polynomial is closely
related with the Chebyshev polynomials of first and second kind. In addition,
we give explicit formulas for the characteristic polynomial of the Coxeter
adjacency matrix, we compute the associated polynomials and use them to derive
the Coxeter polynomial of the underlying graph. We determine the expression of
the Coxeter and associated polynomials as a product of cyclotomic factors. We
use this data to propose an algorithm for factoring Chebyshev polynomials over
the integers. Finally, we prove an interesting formula which involves products
of sines, the exponents, the Coxeter number and the determinant of the Cartan
matrix.
