  This article is the first part of a series of three articles about compatible
systems of symplectic Galois representations and applications to the inverse
Galois problem.
  In this first part, we determine the smallest field over which the
projectivisation of a given symplectic group representation satisfying some
natural conditions can be defined. The answer only depends on inner twists. We
apply this to the residual representations of a compatible system of symplectic
Galois representations satisfying some mild hypothesis and obtain precise
information on their projective images for almost all members of the system,
under the assumption of huge residual images, by which we mean that a
symplectic group of full dimension over the prime field is contained up to
conjugation. Finally, we obtain an application to the inverse Galois problem.
