Let $\mathfrak{g}$ be a finite dimensional, simple, complex Lie algebra. Define the Coxeter adjacency matrix to be the matrix $A=2I-C$ where $C$ is the Cartan matrix of $\mathfrak{g}$. Let $a_n(x)$ be the characteristic polynomial of $A$, where $n$ is the size of $A$. Then the roots of $a_n(x)$ are \begin{equation} 2 \cos \frac{ m_i \pi}{h} \end{equation}
where $m_i$ are the exponents of $\mathfrak{g}$ and $h$ is the Coxeter number of $\mathfrak{g}$. Do you know where this result appeared first? I would also like to know if there is a proof of this fact which is not a case by case verification. I believe that the first such proof is via the Coxeter polymomial whose roots are well-known. I have computed $a_n(x)$ for all simple, complex Lie algebras in http://arxiv.org/abs/1110.6620 but that is a case by case computation.
