@dougmerritt - the vector cross product in 7 dimensions is a very mysterious thing, but luckily it's just a spinoff of the octonions: to the extent we understand the octonions we understand it. I've been thinking about it for a long time. I thought I understood it pretty well.
The thing that's interesting me now - the thing I learned just yesterday! - is that just as 3d rotations act on 3d vectors in a way that preserves their dot product and cross product, 3d rotations also act on 7d vectors in a way that preserves their dot product and cross product. That's bizarre.
The "irreducibility" business is a way of saying that we're not getting this to happen using a cheap trick. If we dropped the irreducibility condition, we could think of 3d rotations as 7d rotations that just happen to only mess around with 3 of the coordinates. We are not doing that here!
So this is weird. By the way, in general, 7d rotations DON'T act on 7d vectors in a way that preserves their dot product and cross product. Only certain special ones do.
(It takes 28 numbers to specify a general rotation in 7 dimensions, and they all preserved the dot product of 7d vectors. Far fewer preserve the cross product too: those can be specified using only 14 numbers.)