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The truth is that the RH Rule is built into the Clifford algebra.
An easy example is with the Pauli matrices, usually defined as:
(read s_{x} as "sigma sub-x")
s_{x}s_{y}=is_{z} s_{i} squared =1, s_{i}s_{j}=-s_{j}s_{i},i not equal j
This corresponds to a RH- coordinate system. One could just as
well have written:
s_{y}s_{x}=is_{z}
which would correspond to a LH-coordinate system.