This identity is used when solving the Dirac equation in Peskin & Schroeder and other texts:
$$(p\cdot \sigma)(p\cdot \bar \sigma)=p^2=m^2 \tag{3.51},$$
and although it seems simple enough I cannot for the life of me arrive at the answer. This is as close as I can get:
$$(p\cdot \sigma)(p\cdot \bar \sigma)=(p^0+p^i\sigma_i)(p^0-p^j\sigma_j)=(p^0)^2-p^0p^j\sigma_j+p^i\sigma_ip^0-p^ip^j\sigma_i\sigma_j, \tag{1}$$
From here the second and third term cancel, and we use the standard Pauli matrix identity on the fourth term:
$$\sigma_i\sigma_j=\delta_{ij}\Bbb I+i\epsilon_{ijk}\sigma^k, \tag{2}$$
which gives:
$$(p^0)^2-p^ip^j\sigma_i\sigma_j=E^2-p^ip^j(\delta_{ij}\Bbb I+i\epsilon_{ijk}\sigma^k)=E^2-\vec p^2+ip^ip^j\epsilon_{ijk}\sigma^k=m^2+ip^ip^j\epsilon_{ijk}\sigma^k \tag{3}.$$
From here I need the last term to vanish in order to obtain the correct answer, but I cannot see any way to do so. Could the solutions perhaps be:
$$i\epsilon_{kij}p^ip^j\sigma^k=i(\vec p\times \vec p)_k\sigma^k=0? \tag{4}$$
Unfortunately I thought of this as I was typing the question, but just in case this isn't valid and/or this question is helpful to other people I will still post it.
