Commutator with Pauli spin matrices and the momentum operator

How is $\left[\vec\sigma \cdot \vec p, \vec \sigma \right]$ proportional to $\vec \sigma\times \vec p$, where $\sigma$ are the Pauli spin matrices and $p$ is the momentum operator?

• What part are you having trouble with? – Brian Moths Mar 20 '14 at 5:37

$[\vec\sigma\cdot \vec p, \vec \sigma]_i = [\sigma_j p^j, \sigma_i] = [\sigma_j , \sigma_i]p^j = 2i\epsilon_{jik} \sigma^k p^j= 2i\epsilon_{ikj} \sigma^k p^j =2i(\vec{\sigma} \times \vec{p})_i$.
So $[\vec\sigma\cdot \vec p, \vec \sigma] = 2i(\vec{\sigma} \times \vec{p}).$