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He defines the centre of mass-energy of a system of particles as:

$$\vec{R} = \frac{1}{E} \sum E_i \vec{r_i}$$

And gives that the total momentum of the system is:

$$\vec{P} = \frac{E}{c^2} \frac{d \vec{R}}{dt}$$

In a footnote he says that the proof of this is non-trivial and refers to a couple of papers.

I took a look at this for two particles and got:

$$\frac{d \vec{R}}{dt} = \frac{1}{(E_1 + E_2)^2}[E_1^2 \vec{u_1} + E_2^2 \vec{u_2} + E_1E_2(\vec{u_1} + \vec {u_2}) + (E_1\frac{dE_2}{dt} - E_2\frac{dE_1}{dt})(\vec{r_2} - \vec{r_1})]$$

$$= \frac{1}{(E_1 + E_2)^2}[(E_1 + E_2)(E_1 \vec{u_1} + E_2 \vec{u_2}) + (E_1\frac{dE_2}{dt} - E_2\frac{dE_1}{dt})(\vec{r_2} - \vec{r_1})]$$

$$= \frac{c^2}{E} \vec{P} + \frac{1}{E^2}[(E_1\frac{dE_2}{dt} - E_2\frac{dE_1}{dt})(\vec{r_2} - \vec{r_1})]$$

The total momentum came out but I can't see how the cross terms in the position vectors disappear. I couldn't find any specific references to this online. Any ideas about what's wrong?