Important 4D spaces:
Simulate it. Just simulate it.
The complex dot product is defined as:
E.g. in :
Just like the usual dot product, this will be a positive definite symmetric bilinear form by definition.
with extra structure added to make it into a metric space.
The identity matrix.
Each elliptic space can be modelled with a real projective space. The best thing is to just start thinking about the real projective plane.