As pointed out in that aside, the general vector-bivector multiplication isn't needed to evaluate the reflection case, but I agree it's a little unsatisfying to

As pointed out in that aside, the general vector-bivector multiplication isn't needed to evaluate the reflection case, but I agree it's a little unsatisfying to sweep that under the rug. You can work everything out by breaking it down to the unit basis vectors, for which the products are explicit. One way of writing it, which feels more comfortable to me possibly at the expense of being the "wrong" sort of intuition, is u(v^w) = (u.v)w - v(u.w) + u^v^w (using . for inner product and ^ for outer product). I guess you can also think of the first two terms as the rule for taking the inner product of a vector and a wedge product, though I haven't thought through this completely. Because of the repeated vector in the reflection formula, that last trivector vanishes, so the result is just a vector, as it will be whenever the vector is coplanar with the bivector.