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 is zero, is called a pure quaternion. The square of pure quaternions is always real and not positive.

Multiplication of pure quaternions leads to

Failed to parse (unknown function "\cross"): (0,\vec{x})(0,\vec{y})=(-\vec{x}\cdot\vec{y},\vec{x}\cross\vec{y})

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

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