Factorization of Curvature Operators

Let V be a real finite-dimensional vector space with inner product and let R be a curvature operator, i.e., a symmetric linear map of the bivector space <!-- MATH $\Lambda {\,^2}V$ --> into itself. Necessary and sufficient conditions are given for R to admit factorization as <!-- MATH $R\, = \,\Lambda {\,^2}L$ --> , with L a symmetric linear map of V into itself. This yields a new characterization of Riemannian manifolds that admit local isometric embedding as hypersurfaces of Euclidean space.