  Mumford constructed a family of abelian fourfolds with special stucture not
characterized by endomorphism ring. Galluzzi showed that the weight 2 Hodge
structure of such a variety decomposes into Hodge substructures via the action
of Mumford-Tate group, one of which is of K3 type with Hodge number (1,7,1). We
will compute the intersection form of such a Hodge structure and provide a
canonical integral Hodge structure on this subspace. Furthermore, we shall show
two applications of this invariant. Firstly, using the above formula we shall
show that every K3 surface obtained in this way will admit an elliptic
fibration. Comparing them with the list by Shimada, we will show that the
Mordell-Weil group of all elliptic fibrations have either 2-torsion or trivial
torsion group, and the conditions of the occurance of 2-torsions can be
specified. Secondly, we shall use it to determine the quaternion that gives a
special CM abelian fourfold that is derived equivalent to the Shioda fourfold.
