Given an integral positive-definite rank $n$ quadratic form $f$, one can use the algorithm in Conway and Sloane (Chapter 15, [SPLaG](https://link.springer.com/book/10.1007/978-1-4757-6568-7)) to efficiently determine if the genus of $f$ contains more than one spinor genus. My question is: given two (integral positive-definite) forms $f,g$ in the same genus, such that the genus contains more than one spinor genus, can one efficiently determine if the forms lie in distinct spinor genera?


I am interested in $n\geq5$.



Computing spinor equivalence for positive definite forms
Yes: one can write down the explicit local transformations at all primes where they are not both unimodular and evaluate the spinor norms and the local automorphism groups. Magma in fact claims to implement such an algorithm, but I don't have personal experience with it.


