Think of a non-singular quadratic form as identifying the vector space with its dual. Given two, you can compose to get a diagonalisable automorphism of the vector space. The intersection of the orthogonal groups centralises that automorphism. To put it another way, you can simultaneously diagonalise the two forms.

Each eigenspace of the automorphism is preserved by the intersection, and the two forms are proportional there. Thus the intersection is a direct product of orthogonal groups acting on the eigenspaces of the automorphism.

(These comments hold for arbitrary values of 3)

Think of a non-singular quadratic form as identifying the vector space with its dual. Given two, you can compose to get an automorphism of the vector space. The intersection of the orthogonal groups centralises that automorphism. As long as this automorphism is diagonalisable, you can simultaneously diagonalise the two forms. This occurs if, for example, one of them is positive definite or negative definite.

In this case, each eigenspace of the automorphism is preserved by the intersection, and the two forms are proportional there. Thus the intersection is a direct product of orthogonal groups acting on the eigenspaces of the automorphism.

If the automorphism is not diagonalisable, the centraliser is smaller and more complicated, but is still the intersection of the two orthogonal groups.

(These comments hold for arbitrary values of 3)