Chapter One Basic Definitions Before we can give the definition of a quadrangular algebra (in 1.17 below), we need to review a few standard notions. Definition 1.1. A quadratic space is a triple (K, L, q), where K is a (com- mutative) field, L is a vector space over K and q is a quadratic form on L, that is, a map from L to K such that (i) q(u + v) = q(u) + q(v) + f (u, v) and (ii) q(tu) = t 2 q(u) for all u, v L and all t K, where f is a bilinear form on L (i.e. a symmetric bilinear map from L × L to K). A quadratic space (K, L, q) is called anisotropic if q(u) = 0 if and only if u = 0. A basepoint of a quadratic space (K, L, q) is an element 1 of L such that