Conservation of 4-momentum in special relativity

I understand that the inner product of two 4-vectors is conserved under the Lorentz transformations, so that the absolute value of the four momentum is the same in any reference frame. This is what I (most likely mistakenly) thought was meant by the conservation of momentum. I don't understand why equations such as

$P_1=P_2+P_3$

($P_i$ are 4-momentum vectors for different particles in a collision for example)

should hold, within a reference frame. I've been told that you can't just add four velocities together on collision of particles, so why should you be able to do this with the momentum vectors?

Answer

I understand that the inner product of two 4-vectors is conserved under the Lorentz transformations

Yes, $p_1.p_2$ is a Lorentz invariant

So that the absolute value of the four momentum is the same in any reference frame.

It is not correct to speak about the "absolute value" of a (quadri)vector. Which is conserved in a Lorentz transformation is $p^2 = (p^o)^2 - \vec p^2$

This is what I (most likely mistakenly) thought was meant by the conservation of momentum.

No, conservation of momentum is a completely different thing. Ultimately, you have some theory describing fields and interactions, describing by an action which is invariant by some symmetries. If the action is invariant by space and time translations, then there is a conserved quantity which is momentum/energy.

I don't understand why equations such as P 1 =P 2 +P 3 (P i are 4-momentum vectors for different particles in a collision for example) should hold, within a reference frame. I've been told that you can't just add four velocities together on collision of particles, so why should you be able to do this with the momentum vectors?

If the theory action is invariant by space/time translations, then the momentum/energy is conserved, so the total momentum/energy of the initial particles is the same as the total momentum/energy of the final particles :

$$(p_\textrm{tot})_\textrm{in}^\mu = (p_\textrm{tot})_\textrm{out}^\mu\tag{1}$$

If there are several initial particles, they are considered as independent (the global state is the tensor product of the states of the initial particles). The independence means that you have :

$$(p_\textrm{tot})_\textrm{in}^\mu = \sum_i p_i^\mu\tag{2}$$ where the sum is about all the initial particles. A similar equation holds for the final particles.