  Mathematical instanton bundles of rank 4 and $c_2=2$ on ${\mathbb P}^4$ have
a smoothquasiprojective moduli space, which is shown via a direct GIT
construction. A complete classification of jumping lines of these vector
bundles is obtained. The duals of these bundles are described. Dimension
computations and irreducibility proofs for some "interesting" subsets of the
moduli space are presented. Restrictions of vector bundles from ${\mathbb P}^4$
to ${\mathbb P}^3$ are shown to produce mathematical instanton bundles on
${\mathbb P}^3$.
