This is just a reference request on a relatively elementary level (for which I apologize in advance), but every time I bump into this question I suspect I'm missing the "correct" conceptual setting. In the simplest case, one is given two vector spaces $V, W$ over a field of characteristic 0, each endowed with a symmetric bilinear form. Then the tensor product $V \otimes W$ inherits an obvious symmetric bilinear form. A natural result is that nondegeneracy of the given forms implies nondegeneracy of the new form, though the proof seems to require somewhat messy manipulation of bases and indices. Even if the vector spaces are infinite dimensional, the same principle seems valid. Then there is the possibility of working over a field of prime characteristic, as well as passing to free modules over commutative rings, etc. 



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> Where in the textbook literature can one find the most definitive treatment of nondegeneracy of symmetric bilinear forms on tensor products? (Preferably with few indices to keep track of.) 
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Optimal reference for tensor product of symmetric bilinear forms?
A rather satisfactory treatment (at least in my opinion) can be found in Greub's book "Multilinear Algebra". 


In the copy I have (the 1967 edition) the nondegeneracy of bilinear forms on tensor products is considered in Chapter I, Section 7, Subsection 1.22 (p. 31).


The arguments involved are index-free and work in any characteristic.


