Consider the conjugate action of on : .

1. This action factors through an action of .

2. There exists a 3D invariant subspace .

3. The determinant is an invariant quadratic form on , and the signature of this form is .

Let be a quadratic form on , whose isometry group is , where .

This induces an injection , and an identification between and the connected component of .

The action on passes on to the projective space . The cone is invariant, and separates into two domains: one of them is homeomorphic to a disk, which the other is a Mobius band. This induces an action of on the disk.

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