ABSTRACT
We show here that every nonsingular symmetric space can be decomposed essentially in a unique way as the direct orthogonal sum (DOS) of two fundamental types of nonsingular symmetric spaces. These two fundamental types are metabolic and anisotropic spaces. While existence of a decomposition is easily proved, its uniqueness in the general case can be established only after a careful analysis of the structure of the space. In case of fields of characteristic ≠ 2 this is much simpler and relays on Witt’s cancellation theorem. The reader interested only in the case when characteristic of the ground field is ≠ 2 will find everything needed in the first section and can safely skip rest of the chapter. However, in §§12.2–12.3 we do not discuss exclusively the case of characteristic two. Instead we give uniform arguments good for all fields and establish deeper properties of totally isotropic subspaces so that it is advisable to have a look at these.
