Orthogonal IAPS

Definition
The orthogonal IAPS associated to a commutative unital ring $$R$$ (usually a field) is the IAPS of groups defined as follows:


 * Its n^{th} member is the orthogonal group $$O(n,R)$$: the group of $$n \times n$$ matrices $$A$$ such that $$AA^t$$ is the identity matrix
 * Its block concatenation map $$\Phi_{m,n}$$ is described as follows:

$$\Phi_{m,n}(A,B) = \begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix}$$

Inside the GL IAPS
The orthogonal IAPS is a sub-IAPS of the GL IAPS, which comprises the general linear groups. It is in fact a saturated sub-IAPS, and the quotient space can be identified with diagonalizable bilinear forms.