This map restricts to a map whose domain is S 2 and, since it is a purely quadratic polynomial, it can be factorised to give a map P 2 (R) → R 4.
This map restricts to a map whose domain is S 2 and, since each component is a homogeneous polynomial of even degree, it takes the same values in R 4 on each of any two antipodal points on S 2. This yields a map P 2 (R) → R 4.