Spin groups over a commutative ring and the associated root data (odd rank case).
Jordan pairs of quadratic forms with values in invertible modules.
Jordan pairs of quadratic forms are generalized so that they have forms with values in invertible modules. The role of such pairs turns out to be natural in describing 'big cells', a kind of open charts around unit sections, of Clifford and orthogonal groups as group schemes. Group germ structures on big cells are particularly interested in and related also to Cayley-Lipschitz transforms.
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