The dual coxeter number is the inverse of the norm of the highest root: looking for a simple proof Does anybody know a simple generic proof (or a reference for it) for the following formula?:
If $\theta=\sum a_i\alpha_i$ is the highest root of a simple complex lie algebra with $\Delta=\{\alpha_1, \cdots,\alpha_n\}$ as simple roots system, we denote $\theta^{\vee}=\sum c_i\alpha_i^{\vee}$ the corresponding coroot.
The dual Coxeter number is then defined as $h^{\vee}=1+\sum c_i$, and we have the formula for which I am looking for a proof
$$h^{\vee}=\frac1{\kappa^*(\theta, \theta)}$$
where $\kappa^*$ is the scalar product on the dual of the Cartan subalgebra of the Lie algebra coming from the Killing form.
 A: Looking at a seminal paper of Dynkin, I found the following proof that I include here with simplified notations, with the algebra of rank $r$:
We use a Cartan-Weyl basis $((H_i), (E_\alpha))$ where $1\leq i\leq r$, $(H_i)$ an orthonormal basis for the Killing form of the Cartan subalgebra, and $(E_\alpha)$ the root vectors belonging to the root spaces of the algebra, with $\alpha\in\Phi$, the set of all roots, and normalize such that $\kappa(E_\alpha, E_{-\alpha})=1$.
The Casimir operator is
$$C=\sum_i H^2_i+\sum_{\alpha\in\Phi^+}(E_\alpha E_{-\alpha}+E_{-\alpha} E_{\alpha})$$
By checking its trace, it is easy to see that it is the identity on the adjoint representation of the algebra.
If $\theta$ is the highest root, and $E_\theta$ a non zero vector of its root space, which is a highest weight vector for the adjoint representation, we have $\alpha\in\Phi^+$ $\text{ad}(E_\alpha)(E_\theta)=0$, therefore
$$\text{ad}(C)(E_\theta)=\kappa(E_\theta, E_\theta)E_\theta+\sum_{\alpha\in\Phi^+}\kappa(E_\theta, E_\alpha)E_\theta=(\kappa^*(\theta, \theta)+2\kappa^*(\theta, \rho))E_\theta$$
where $\rho$ is the half-sum of all the positive roots.
We get immediately
$$1=\kappa^*(\theta, \theta)h^{\vee}$$
