I tried posting this to the Math Message Board, but I'm not getting any responses there...
I don't use the Clifford, or Geometric product, here, so when there's no operator I mean apply ordinary multiplication of scalars.
I found this rather easy way to resolve wedge products here... wedge-product (This is why I sometimes drop explicitly stating the basis vectors) The author's method appears to work, and he doesn't use the basis vectors explicitly... however, if the subscripts of the vector components themselves stand for the basis vectors, then having e<sub>31</sub> be the <i>default</i>? basis bivector would mess his system up,... or at least, his system loses it's correspondence to the basis vectors...?
I thought Hamilton choose the basis bivectors for the quaternions because he decided he wanted a left-handed basis... which makes ijk = -1, as he desired. However, he also has the negative signs, which is equivalent to one additional transposition...?
I'm confused... how do I justify using e<sub>31</sub>, instead of e<sub>13</sub>...?
Can someone go over this is some detail... it seems like this could be of critical importance. How do I know how to choose the permutation ordering of basis bivectors, trivectors, etc. if I go to 4, or more, dimensions?