  In this paper we classify real trivectors in dimension 9. The corresponding
classification over the field C of complex numbers was done by Vinberg and
Elashvili in 1978. One of the main tools used for their classification was the
construction of the representation of SL(9,C) on the space of complex
trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of
the simple complex Lie algebra of type E8. This divides the trivectors into
three groups: nilpotent, semisimple, and mixed trivectors. Our classification
follows the same pattern. We use Galois cohomology to obtain the classification
over R. For the nilpotent orbits this is in principle rather straightforward
and the main problem is to determine the first Galois cohomology sets of a long
list of centralizers (we compute the centralizers using computer). For the
semisimple and mixed orbits we develop new methods based on Galois cohomology,
first and second. We also consider real theta-representations in general, and
derive a number of results that are useful for the classification of their
orbits.
