G is a 4th dimensional vector space over R so it's underlying set is not R^2. The geometric product of two vectors is not an element of R^2 - it's a real numbe

G is a 4th dimensional vector space over R so it's underlying set is not R^2. The geometric product of two vectors is not an element of R^2 - it's a real number + a bivector. We are really working in the exterior algebra.