I read all of this and still can't fathom which of these coordinate system types is this:
(0,0,1) | | * P | / \ / \ / \ (1,0,0) (0,1,0)
Any point P is reachable by linear combination of basis vectors. Only positive scalars required (vector reversal operation not required to span plane, only grow/shrink and tip-to-tail addition).
The 4 basis vector analog of the above is what I've dubbed "quadrays" and describe at my website at http://www.teleport.com/~pdx4d/quadrays.html -- 4 vectors from the origin thru the vertices of a regular tetrahedral enclosure. All points in volume uniquely identified by 4-tuples with no negative scalars.
 we can define negative scalars and include vector reversal i.e. multiplication of a vector by -1, but these more provide redundant shortcuts in notation than trully novel operations.