  This paper is devoted to the study of Poisson structures on the Euclidean
four dimensional space R4. By using the properties of the trace operator
associated to a volumen form and the elementary vector calculus operations in
R3, we give explicit formulas for the main geometric objects associated to the
Poisson structures in R4, including its characteristic foliation, the
Hamiltonian and Poisson vector fields, normal forms and some useful
decomposition formulae for Poisson tensors. We also discuss the class of
unimodular Poisson structures and give two results about the existence of
Poisson structures having as its characteristic foliation a given arbitrary
regular foliation.
