Over a base scheme, I will discuss a class of quadratic forms that have the simplest type of nontrivial degeneration along a divisor. Such forms naturally arise in number theory and algebraic geometry; I will give examples related to Gauss composition and to cubic fourfolds containing a plane. Quadratic forms with such simple degeneration turn out to be torsors for orthogonal group schemes that are smooth, yet not reductive, over the base. I will describe the local structure of these orthogonal group schemes, which are interesting objects in their own right.