Geometric algebra, like Clifford algebra, derives its structure from the geometric product on a vector space, and keeping track of the grades of its parts. The geometric semantics which distinguishes geometric
algebra from Clifford algebra is exposed by considering various `derived' products. Prime among those is the outer product, encoding the Grassmann algebra of spanning subspaces. Secondary in standard
literature is the inner product, related to metric perpendicularity. Yet the standard inner product does not quite mesh with the outer product, leading to many identities which are conditional on the grades of their
arguments.

In this paper, we instead `split' the geometric product using the outer product and the scalar product. The latter is very fundamental (more so than we had realized), since it transfers the concept of metric to blades.
We show how this approach leads naturally to a slightly different inner product, the contraction. Identities expressed in terms of the contraction are not conditional on grades for their validity, and we believe this makes them easier to use in geometric programming. The geometrical meaning of the contraction is similar to
that of the standard inner product, but it now more tidily codifies properties both of the perpendicularity of subspaces, and of their mutual containment.

We then show how the contraction simplifies the formulation and treatment of important operations such as projection, meet and join. We also derive the conditions under which expressions involving inner products are meaningful, in the sense of transforming covariantly under linear transformations. Our conclusion is that the use of the contraction rather than the inner product improves the coherence and algebraic structure of
geometric algebra, without sacrificing any of its geometric significance.

leo@science.uva.nl