The "-handed" labels are a reflection of terminology in 3D geometry and physics - a system of 3D axes is termed "Left-handed" if these axes, in ascending order of index (e.g. x,y,z or i,j,k or e1, e2, e3) have the same orientation as the thumb, index finger, and middle finger of the left hand; similarly for "Right-handed", in mirror-image. Parity and chirality are concepts in physics intimately related to this notion of handedness. Note that a transformation of one table via axis inversion will give the other table. The same convention defining handedness (i.e. a triad of basis elements (ei, ej, ek) i < j < k is "Right-handed" if eiej = +ek, and "Left-handed" if eiej = -ek) will be used to define a more complex notion of parity in higher-dimensions, where axis inversion is not equivalent to handedness reversal.|
One further observation that will be helpful later - the indexes i, j, k in the above multiplication tables obey
eiej = [+/-] ei^j (modulo sign),
where '^' denotes bitwise exclusive OR (XOR).