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?



A formula for the dual Coxeter number
By Weyl invariance, it suffices to prove this for *some* long root. When $\alpha\_0$ is the highest root, it is an immediate consequence of Lemma 4 of [Suter - Coxeter and dual Coxeter numbers](https://doi.org/10.1080/00927879808826122), which states that $h^\vee\alpha\_0 = \sum\_{\alpha \in \Phi\_+} \langle\alpha, \alpha\_0^\vee\rangle\alpha$.


