On the Conjugacy of Orthogonal Groups

In much of the mathematical literature, there is talk about the (note the definite article) orthogonal group O(n, F ) of degree n ∈ NI over a field F . This is extremely misleading, because, given a linear space T of dimension n, one can consider the orthogonal group of any non-degenerate quadratic form Q on T . In Chapter 6 of the book Basic Algebra I [J], Jacobson denotes this orthogonal group by O(Q). Since there are many such quadratic forms, there are many orthogonal groups. The main purpose of this paper is to study the relations between them and, in particular, determine under what conditions they are conjugate, I was able to find a complete answer in the case when F is an ordered field.