Splitting Fields of Characteristic Polynomials in Algebraic Groups

In [K1] and earlier in [K2], questions of the following type are considered: suppose a family (gi)i of matrices in some (algebraic) matrix group are given, with rational coefficients. What is the “typical” Galois group of the splitting field Ki of the characteristic polynomial of gi (defined as the field generated over Q by the roots of the characteristic polynomial)? Is this characteristic polynomial typically irreducible? If the elements gi are in GL(n,Q), or in SL(n,Q), there is an obvious “upper bound”, namely the symmetric group Sn. If the elements gi are in a symplectic group (for an alternating form with rational coefficients), or in the group of symplectic similitude, there is also an easy, if slightly less obvious, upper bound: the characteristic polynomial satisfies some relation such as