An Introduction to Geometric Algebra
Please don't conflate sets with algebras. Strictly speaking the tuple (R^2, +, ·) (where · is multiplication by a scalar) is a linear algebra over R, R^2 is just a set.Now I know mathematicians often refer to spaces by their underlying set, but the notation G^2 doesn't really make sense (for starters R^2 = R x R, what is G?). Calling it the Geometric Algebra over R^2 would be fine. You can even refer to it as just R^2, as long it's clear that you're using the geometric product (and what metric you're using).