  In the 80's Aschbacher classified the maximal subgroups of almost all of the
finite almost simple classical groups. Essentially, this classification divide
these subgroups into two types. The first of these consist roughly of subgroups
that preserve some kind of geometric structure, so they are commonly called
subgroups of geometric type. In this paper we will prove the existence of
infinitely many compatible systems $\{ \rho_\ell \}_\ell$ of $n$-dimensional
Galois representations associated to regular algebraic, essentially self-dual,
cuspidal automorphic representations of $\mbox{GL}_n(\mathbb{A}_{\mathbb{Q}})$
($n$ even) such that, for almost all primes $\ell$, the image of
$\overline{\rho}_{\ell}$ (the semi-simplification of the reduction of
$\rho_\ell$) cannot be contained in a maximal subgroup of geometric type of an
$n$-dimensional symplectic or orthogonal group. Then, we apply this result to
some 12-dimensional representations to give heuristic evidence towards the
inverse Galois problem for even-dimensional orthogonal groups.
