Let $R$ be one of the $A,B,C,D,E,F$ root systems, let $\Lambda$ be the lattice generated by the roots, $R^+$ a system of poitive roots and $\rho$ and $\alpha$ be the Weyl vector and the highest root corresponding to $R^+$. Let $h$ be the real number such that $h(x,x)=\sum_{r\in R^+} (x,r)^2$ for all $x$ in the ambient Euclidean space (where $(\cdot,\cdot)$ denotes the scalar product).

I needed the following characterisation of the Weyl vector.

Lemma. Suppose $v\in \rho+\Lambda$, $(v,r)>0$ for all positive roots, $h>(v,\alpha)$. Then $v=\rho$.

I checked this lemma by a case by case analysis using the description of the root systems as in Bourbaki's "Groupes et algebres de Lie". (To be honest, I did not yet check $E_6,E_7$).

My question: Is that lemma known in this or another disguise, or can someone hint me to a more conceptual and instructive proof than that explicit checking of each root system?