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Perhaps I seem to be repeating myself as you are repeatedly ignoring what I'm writing. The tables are unclear as the order of the product is unclear from them. E.g. the top-right corner of the first table is e6. But does that mean e1 × e7 = e6 or e7 × e1 = e6 ? It would not matter if the product is commutative but as it's not the tables are ambiguous. That is why tables are a very poor way to show the product, and probably why none of the sources use them.
And again please read MOS:LEAD in paticular the section on providing an accessible overview. Or look at the articles already mentioned, Special relativity and Cross product. Or one you are familiar with, Pythagorean theorem. Or a similarly advanced article to this one (and a featured article), Laplace–Runge–Lenz vector. There should be little or no maths in the introduction: perhaps a statement of the product, but nothing else. Certainly not the page of detailed mathematical working and argument that there is now.wordsdeeds "