- there are 4 basis elements:
- e
_{0}(equivalent to the unit element of the real numbers), and - 3 other distinct "imaginary" elements e
_{1}, e_{2}, e_{3}such that e_{i}^{2}= -1

(historically these imaginary elements were first called i, j, k)

- e
- the basis elements multiply according to one of the two following "multiplication tables":
e _{0}e _{1}e _{2}e _{3}**e**_{0}e _{0}e _{1}e _{2}e _{3}**e**_{1}e _{1}-e _{0}-e _{3}e _{2}**e**_{2}e _{2}e _{3}-e _{0}-e _{1}**e**_{3}e _{3}-e _{2}e _{1}-e _{0}OR e _{0}e _{1}e _{2}e _{3}**e**_{0}e _{0}e _{1}e _{2}e _{3}**e**_{1}e _{1}-e _{0}e _{3}-e _{2}**e**_{2}e _{2}-e _{3}-e _{0}e _{1}**e**_{3}e _{3}e _{2}-e _{1}-e _{0}(the interrelationship between these two alternatives will be discussed below, and is important in generalizing to higher dimensions) "Left-handed" "Right-handed" i ^{2}=j^{2}=k^{2}=kji=-1i ^{2}=j^{2}=k^{2}=ijk=-1

- an arbitrary quaternion is of the form r
_{0}e_{0}+ r_{1}e_{1}+ r_{2}e_{2}+ r_{3}e_{3}, with r_{i}real numbers - the real coefficients r
_{i}are commutative and associative with any multiplicative combination of the basis elements,

*e.g.*r_{i}e_{i}((r_{j}e_{j})(r_{k}e_{k})) = r_{i}r_{j}r_{k}(e_{i}(e_{j}e_{k})) - multiplication is distributive over addition

Restricting attention to just the set {+e

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 e_{1}, e_{2}, e_{3}) 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 (e_{i}, e_{j}, e_{k}) i < j < k is "Right-handed" if e_{i}e_{j} = +e_{k}, and "Left-handed" if e_{i}e_{j} = -e_{k}) 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 e _{i}e_{j} = [+/-] e_{i^j} (modulo sign),where '^' denotes bitwise exclusive OR (XOR). |