In Eric Lengyel’s article “Projective Geometric Algebra Done Right” (Projective Geometric Algebra Done Right – Eric Lengyel's Blog) he introduces a new product, the antidot product. He briefly mentions the symmetry between the dot and antidot product with the following quote “The antidot product works with antivectors in the same way that the dot product works with vectors, but it produces an antiscalar quantity instead of a scalar quantity.”

How exactly is this product defined? I referenced Eric’s Cayley table (http://projectivegeometricalgebra.org/projgeomalg.pdf) and can’t understand how the geometric antiproduct (which includes the antidot product) between the vector e4 and itself can result in the negative antiscalar. All the products in the table that resolve to just the antidot products are just as opaque.

Any help is much appreciated.