In previous joint work with J.-S. Koskivirta, we introduced the notion of quasi-constant character (of a maximal torus of a connected reductive group over a field); we showed that over an algebraically closed field it naturally unifies the notions minuscule and co-minuscule. In this note we characterize quasi-constant fundamental weights in terms of the Weyl group of the corresponding maximal Levi subgroup. Equivalently, purely in the language of root systems, the result characterizes special and co-special vertices of Dynkin diagrams in terms of the Weyl group of the corresponding maximal sub-root system.