Let \(\vec{\sigma}=\left(\sigma_1, \sigma_2, \sigma_3\right)\), where \(\sigma_1, \sigma_2, \sigma_3\) are the Pauli matrices. If \(\vec{a}\) and \(\vec{b}\) are two arbitrary constant vectors in three dimensions, the commutator \([\vec{a} . \vec{\sigma}, \vec{b} . \vec{\sigma}]\) is equal to (in the following I is the identity matrix)
1
\((\vec{a} . \vec{b})\left(\sigma_1+\sigma_2+\sigma_3\right)\)
2
\(2 i(\vec{a} \times \vec{b}) \cdot \vec{\sigma}\)
3
\((\vec{a} . \vec{b}) I\)
4
\(|\vec{a}||\vec{b}| I\)