Pure and unit quaternions

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Pure quaternion

A quaternion whose vector part is zero equals a real number corresponding to the scalar part.

A quaternion whose scalar part

$i^2 = j^2 = k^2 = ijk = -1$

Unlike the multiplication of real or complex numbers, the multiplication is not commutative:

$\begin{align} ij &= -ji = k \\ jk &= -kj = i \\ ki &= -ik = j \end{align}$