I had the same reaction as you, but then I skipped to the end where he said that mathematicians call this a Clifford algebra[0], and with that it all made sense. I agree with jacobolus that mathematicians are clearly not the target audience of this paper, so the mathematical soundless is left as an exercise to the reader if they are so inclined.

[0] http://en.wikipedia.org/wiki/Clifford_algebra#Introduction_a...