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Roe's book is one of my favorite books, and the place where I first learned some index theory. I know that the lack of a few details can be frustrating sometimes, though. Let me see if I can address some of your questions. $(,)$ is the Hermitian metric on $S$ that Roe refers to in Definition 3.4. You're right that he's not too explicit about this. (I ...
Qiaochu's comment is certainly the first thing to point out: since quadratic forms and bilinear forms are not interchangeable in characteristic $2$, you need to think about which you really are interested in, at some point. (Do we use $a\otimes a+Q(a)$ as relations or $a\otimes a + B(a,a)$?) Geometrically the characteristic $2$ case is exceptional, but ...