Modules of derivations and semi-simple groups

In analogy with the "module of derivations" in hyperplane arrangement theory, we consider derivations of the coordinate ring of an algebraic torus which are tangent to a codimension one subvariety consisting of a union of finitely many sub-tori. An important example is obtained by taking a maximal torus T in a semi-simple group G with subvariety consisting of the union of the kernels of the roots of G. We show that in this case the module of derivations is a free module over the coordinate ring of T.