Sorry for digging this up after a year, but there's one thing I'm curious about. What kind of vector product are we talking here, exactly? I've been playing a little with the math but I can't get one thing.
If it's cross product, then:
e_1e_1 = 0
And not 1. So:
e_1X = a e_1e_1 + b e_1e_2...