Thanks for the post, I found it very enlightening. But I'm having trouble understanding
"The vector ... will contain a component in the RRR direction of U_{11} U_{12} U_{13}, but all six terms will only differ by a sign, and the sum will be zero."
Isn't the U_{11} U_{12} U_{13} term the only thing contributing to the RRR coefficient? What are the six terms you reference here?