  Let us consider the rank 14 lattice $P=D_4^3\oplus < -2> \oplus < 2>$. We
define a K3 surface S of type P with the property that $P\subset {\rm Pic}(S)
$, where ${\rm Pic}(S) $ indicates the Picard lattice of S. In this article we
study the family of K3 surfaces of type P with a certain fixed
multipolarization. We note the orthogonal complement of P in the K3 lattice
takes the form $$ U(2)\oplus U(2)\oplus (-2I_4). $$ We show the following
results: \item{(1)}
  A K3 surface of type P has a representation as a double cover over ${\bf
P}^1\times {\bf P}^1$ as the following affine form in (s,t,w) space: $$ S=S(x):
w^2=\prod_{k=1}^4 (x_{1}^{(k)}st+x_{2}^{(k)}s+x_{3}^{(k)}t+x_{4}^{(k)}), \
x_k=\pmatrix{x_{1}^{(k)}&x_{2}^{(k)}\cr x_{3}^{(k)}&x_{4}^{(k)}} \in M(2,{\bf
C}). $$ We make explicit description of the Picard lattice and the
transcendental lattice of S(x). \item{(2)}
  We describe the period domain for our family of marked K3 surfaces and
determine the modular group. \par \noindent \item{(3)}
  We describe the differential equation for the period integral of S(x) as a
function of $x\in (GL(2,{\bf C}))^4$. That bocomes to be a certain kind of
hypergeometric one. We determine the rank, the singular locus and the monodromy
group for it. \par \noindent \item{(4)} It appears a family of 8 dimensional
abelian varieties as the family of Kuga-Satake varieties for our K3 surfaces.
The abelian variety is characterized by the property that the endomorphism
algebra contains the Hamilton quarternion field over ${\bf Q}$.
