Paul Schwahn on Nostr: great post, great explanation! I must add, the space of bivectors becomes isomorphic ...

great post, great explanation! I must add, the space of bivectors becomes isomorphic to 𝔰𝔬(p,q), not End(V).

Another comment: you make the argument that multivectors have more geometric intuition than k-forms. But I think this intuition cannot be stretched too far (I'm sure you know all this, but I gotta be pedantic): a (simple) bivector 𝑎∧𝑏 is NOT the parallelogram spanned by 𝑎 and 𝑏, but rather some sort of "oriented amount" of the plane spanned by the two vectors. The parallelogram is only one way to represent 𝑎∧𝑏, but there are many other parallelograms (with same signed area) that represent the same bivector. The bivector 𝑎∧𝑏 does not know anything about the angles and lengths of 𝑎,𝑏.

It becomes much more unintuitive when you ADD different bivectors - and you'll have to, because not all bivectors are simple!

Passing to 2-forms is not very hard conceptually: think of them as being "measuring devices" for bivectors in the sense that they are dual to them. In fact, in order to define a 2-form it suffices to define it on simple bivectors, so in my mind this makes them actually MORE geometrically intuitive than bivectors!

However I strongly agree with the point that we should encourage to identify bivectors, 2-forms and infinitesimal rotation once we have a metric available - this really does help! But as a Riemannian geometer I might be biased here.