  We prove a version of Linnik's basic lemma uniformly over the base field
using theta-series and geometric invariant theory in the spirit of Khayutin's
approach (Duke Math. J., 168(12), 2019). As an application, we establish
entropy bounds for limits of invariant measures on homogeneous toral sets in
GL(4) of biquadratic, cyclic, or dihedral type.
