The way this text does the geometric product is trying to be intuitive by immediately giving a "physical" interpretation as the sum of the inner and outer produ

The way this text does the geometric product is trying to be intuitive by immediately giving a "physical" interpretation as the sum of the inner and outer products. However the physical interpretations breaks down when trying to multiply three of these together. I think Ben Lynn's approach is actually better: https://crypto.stanford.edu/~blynn/haskell/ga.html https://crypto.stanford.edu/~blynn/haskell/ga.html (Part 1) It starts off more abstract, defining the geometric product simply as string concatenation (as in "free monoid" if you're familiar with that term, which you would if you have intermediate Haskell knowledge) plus a very natural constraint. From this natural constraint one can then deduce the sum-based interpretation that the OP gives. The natural constraint is then further generalised and justified in more detail in Part 3: https://crypto.stanford.edu/~blynn/haskell/cga.html https://crypto.stanford.edu/~blynn/haskell/cga.html