  We define a complex relativistic phase space which is the space
$\mathbb{C}^4$ equipped with the Minkowski metric and with a geometric
tri-product on it. The geometric tri-product is similar to the triple product
of the bounded symmetric domain of type IV in Cartan's classification, called
the spin domain. We show that there are to types of tripotents-the basic
elements of the tri-product in the relativistic phase space. We construct a
spectral decomposition for elements of this space. A description of
compatibility of element of the relativistic phase space is given. We show that
the relativistic phase space has two natural bases consisting of compatible
tripotents. The fist one is the natural basis for four-vectors and the second
one is the Newman-Penrose basis. The second one determine Dirac bi-spinors on
the phase space. Thus, the relativistic phase space has similar features to the
quantum mechanical state space.
