[howling]

First of all it is only invertible for some non-zero elements, especially if `a` is a linear combination of multivectors or we work in PGA that explicitly adds a basis vector of norm 0. Yes sometimes it is useful but that doesn't automatically makes it more fundamental than the inner product and exterior product.

[at_compile_time]

You make it sound as though multivectors being invertible is a special case, when the opposite is true. In 2D and 3D GA, every non-zero k-vector and versor has an inverse. In PGA, every non-zero, non-ideal plane, line, and point, and versor has an inverse. The inverse is used all the damn time when composing and applying transformations and performing projections and rejections.

As to which is more fundamental, I don't think it matters. You could argue that the dot and exterior products are more fundamental because the geometric product is their sum (for vectors). You could also argue that the geometric product is more fundamental because it is simply the Cartesian product of two multivectors, and you derive the dot, exterior and commutator products by filtering that product by grade. Both definitions are true, and "fundamental" is both a matter of perspective and irrelevant to any practical concern.

[howling]

> In 2D and 3D GA, every non-zero k-vector and versor has an inverse.

Of course by definition every versor has an inverse. The invertibility of k-vector gets hairier for higher dimensions though. Even in 3D GA, some mixed-grade elements are not invertible.

> As to which is more fundamental, I don't think it matters.

It doesn't matter mathematically but it matters pedagogically. GA enthusiasts seem to advocate teaching GA to anyone that has learnt linear algebra. I believe it is more appropriate to stick to teaching tensor algebra and its quotient exterior algebra. Then it is up to you to learn Clifford algebra as a generalization of exterior algebra; especially if you are a game dev, a physicist, or a topological K-theorist.