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Bill Casselman writes research and expository mathematics connecting algebraic structure with concrete calculations. His author-hosted Introduction to quadratic forms by Bill Casselman explains quadratic forms over general fields, including the distinctions required in characteristic two. It develops decomposition results using totally isotropic subspaces and the hyperbolic plane.^[ref]

For example, over $\mathbb{R}$ the quadratic form $Q(x,y,z)=xy+z^2$ separates into a hyperbolic plane on the first two coordinates and an anisotropic quadratic form on the last. The $x$-axis consists of isotropic vectors, but it is not invisible to the whole form: its associated bilinear form pairs $(1,0,0)$ nontrivially with $(0,1,0)$. Pairing such complementary directions produces the hyperbolic part of an orthogonal direct sum. The example helps distinguish a vector on which a quadratic form vanishes from a vector in the radical of its associated bilinear form.^[ref]

By Bill Casselman.

Verified locator: Author text revised 24 June 2022. Proposition 2.6 and Corollaries 2.7–2.8 give hyperbolic splitting from a totally isotropic subspace; Corollary 6.8 gives uniqueness of the hyperbolic multiplicity and anisotropic complement.