That's my point, you need to remember "ijk" in order (i.e. i=jk or ij=k) in a purely algebraic definition instead of a meaningless and complicated "handedness"

That's my point, you need to remember "ijk" in order (i.e. i=jk or ij=k) in a purely algebraic definition instead of a meaningless and complicated "handedness" rule involving rotations.