Let Μ be a full operator-stable probability measure over a finite dimensional real inner product space V. Necessary and sufficient conditions are obtained for Μ to have independent univariate marginals with respect to some basis of V. These essentially amount to the statement that the support of the Levy spectral measure is a subset of the union of one-dimensional subspaces of V determined by the vectors in a basis of the subspace of V spanned by the support of the non-Gaussian component. A representation for an exponent of such a Μ is also given.