Editing Pauli matrices (section)

=== Cross-product ===

The cross-product is given by the matrix commutator (up to a factor of <math>2i</math>)
<math display="block">
 [\vec a \cdot \vec \sigma, \vec b \cdot \vec \sigma] = 2i\,(\vec a \times \vec b) \cdot \vec \sigma.
</math>
In fact, the existence of a norm follows from the fact that <math>\mathbb{R}^3</math> is a Lie algebra (see [[Killing form]]).

This cross-product can be used to prove the orientation-preserving property of the map above.