Hyperplane arrangements and Vinberg's θ-groups

Let g = ⨁ i ∈ Z / m Z g i be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of g to a Cartan subspace c ⊂ g 1 coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of g = ⨁ i ∈ Z / m Z g i .