Is a subspace of GA(n) closed under the geometric product? Say we let a k-blade represent a subspace of GA(n), where k < n. Does that also represent a subalgebra of GA(n)?
I can see that we won't get any higher dimensional elements. But I'm not sure that the product of 2 blades in this subspace will always stay in the subspace. Can you give me an example of how subspaces are NOT closed under the geometric product? Or if it is, can you point in the direction of how to prove it?
In general, blades representing subspaces do not multiply under the geometric product to produce other blades.
A simple counterexample would be two vectors. All vectors are blades, and any two vectors lie in a plane spanned by those vectors: that plane is formed by the wedge product. But the geometric product of those two vectors will (unless the vectors are orthogonal) produce a scalar term also. The result is not a blade.