  In this paper the Weyl tensor is used to define operators that act on the
space of forms. These operators are shown to have interesting properties and
are used to classify the Weyl tensor, the well known Petrov classification
emerging as a special case. Particularly, in the Euclidean signature this
classification turns out be really simple. Then it is shown that the
integrability condition of maximally isotropic distributions can be described
in terms of the invariance of certain subbundles under the action of these
operators. Here it is also proved a new generalization of the Goldberg-Sachs
theorem, valid in all even dimensions, stating that the existence of an
integrable maximally isotropic distribution imposes restrictions on the optical
matrix. Also the higher-dimensional versions of the self-dual manifolds are
investigated. These topics can shed light on the integrability of Einstein's
equation in higher dimensions.
