Pairs of quadratic forms over a quadratic field extension

Let F be a field of characteristic distinct from 2, L=F(d) a quadratic field extension. Let further f and g be quadratic forms over L considered as polynomials in n variables, Mf, Mg their matrices. We say that the pair (f,g) is a k-pair if there exist S∈GLn(L) such that all the entries of the k×k upper-left corner of the matrices SMfSt and SMgSt are in F. We give certain criteria to determine whether a given pair (f,g) is a k-pair. We consider the transfer corL(t)/F(t) determined by the F(t)-linear map s:L(t)→F(t) with s(1)=0, s(d)=1, and prove that if dimcorL(t)/F(t)(f+tg)an≤2(n−k), then (f,g) is a [k+12]-pair. If, additionally, the form f+tg does not have a totally isotropic subspace of dimension p+1 over L(t), we show that (f,g) is a (k−2p)-pair. In particular, if the form f+tg is anisotropic, and dimcorL(t)/F(t)(f+tg)an≤2(n−k), then (f,g) is a k-pair.