  It is well known that the classification of the Weyl tensor in Lorentzian
manifolds of dimension four, the so called Petrov classification, was a great
tool to the development of general relativity. Using the bivector approach it
is shown in this article a classification for the Weyl tensor in all
four-dimensional manifolds, including all signatures and the complex case, in
an unified and simple way. The important Petrov classification then emerges
just as a particular case in this scheme. The boost weight classification is
also extended here to all signatures as well to complex manifolds. For the Weyl
tensor in four dimensions it is established that this last approach produces a
classification equivalent to the one generated by the bivector method.
