  In a recent series of papers we have shown how the eikonal/geometrical optics
approximation can be used to calculate analytically the fundamental quasinormal
mode frequencies associated with coupled systems of wave equations, which
arise, for instance, in the study of perturbations of black holes in gravity
theories beyond General Relativity. As a continuation to this series, we here
focus on the quasinormal modes of nonrotating black holes in scalar
Gauss-Bonnet gravity assuming a small-coupling expansion. We show that the
axial perturbations are purely tensorial and are described by a modified
Regge-Wheeler equation, while the polar perturbations are of mixed
scalar-tensor character and are described by a system of two coupled wave
equations. When applied to these equations, the eikonal machinery leads to
axial modes that deviate from the general relativistic results at quadratic
order in the Gauss-Bonnet coupling constant. We show that this result is in
agreement with an analysis of unstable circular null orbits around blackholes
in this theory, allowing us to establish the geometrical optics-null geodesic
correspondence for the axial modes. For the polar modes the small-coupling
approximation forces us to consider the ordering between eikonal and
small-coupling perturbative parameters; one of which we show, by explicit
comparison against numerical data, yields the correct identification of the
quasinormal modes of the scalar-tensor coupled system of wave equations. These
corrections lift the general relativistic degeneracy between scalar and
tensorial eikonal quasinormal modes at quadratic order in Gauss-Bonnet coupling
in a way reminiscent of the Zeeman effect. In general, our analytic, eikonal
quasinormal mode frequencies (normalized to the General Relativity ones) agree
with numerical results with an error of $\sim 10\%$ in the regime of small
coupling constant. (abridged)
