Let $\Phi$ be the root system of a finite dimensional simple Lie algebra $\mathfrak g$, with dual Coxeter number $h^\vee$.

Let $\alpha_0\in \Phi$ be a long root

(if all the roots have the same length, then let $\alpha_0\in \Phi$ be any root).

Let $\langle\cdot,\cdot\rangle$ be the basic inner product (the inner product according to which $\langle\alpha_0,\alpha_0\rangle=2$).

Then I believe that the following formula is true: $ \displaystyle\sum_{\alpha\in\Phi}\,\, \langle \alpha,\alpha_0\rangle^2=4h^\vee $

How does one prove that formula?

simpleroots (not all the roots)? That would look more like what I've seen as the definition of the dual Coxeter number. $\endgroup$