  In this paper we give a general construction of transcendental lattices for
K3 surfaces with real multiplication by arbitrary field up to degree 6 along
with formula for their discriminants. We also show that all simple Abelian
fourfolds with definite quaternionic multiplication can be realized as
Kuga-Satake varieties of K3 surfaces with Picard rank 16 and real
multiplication by a quadratic field by keeping track of the arithmetic input on
both sides.
