Is the Equation for Relativistic Two-Body Decay Correct?

[EquationOfMotion]

I was looking up stuff for relativistic two-body decay, and I came across this pdf: https://indico.cern.ch/event/391122/contributions/928962/attachments/782786/1073126/twoBodyDecay.pdf

which says that because $p_2^2=P^2−2P\cdot p_1+p_1^2$ we apparently have $m_2^2=M^2−2ME_1+m_1^2$. But this doesn't seem correct, given that $p_1=(E_1/c,\vec{p_1})$.

Is it correct? If so, how do we get there?

 

[Orodruin]

Yes, it is correct (in the rest frame of P). It always holds that $p_1^2 = m_1^2$.

Note that this is using units of c = 1.