Strict positivity of a nonzero mixed matrix integral

Let $X,Y$ be distinct real $n\times n$ matrices and let $P,Q$ be real symmetric positive-definite matrices. Then the mixed spherical integral with numerator $|u^{\mathsf T}(X-Y)v|$ and denominator $(u^{\mathsf T}Pu)(v^{\mathsf T}Qv)$ is strictly positive:

No separate positive-dimension assumption is needed: the existence of two distinct matrices indexed by $\operatorname{Fin}(n)$ already excludes the zero-dimensional case. This lemma permits cancellation of positive cross-integral factors in denominator-normalization arguments.