In SR, I understand you can use 4 momentum conservation, but what are the special cases where you can use 3 momentum/energy conservation?

An example I have seen is with $$P_1=(M_1, 0), P_2=(M_2,0),P_3=(\sqrt{(m_3^2+p^2)},p), P_4 =(|p|,-p)$$

where P1 is the initial state and P2,P3,P4 are the final. From here the example said

$$M_1 = M_2 + \sqrt{(m_3^2+p^2)} + |p|$$

Could somebody explain why we are allowed to say that, I thought that energy and momentum became "intertwined" and energy and momentum conversation were coupled into four momentum conservation.

In SR, I understand you can use 4 momentum conservation, but what are the special cases where you can use 3 momentum/energy conservation?

An example I have seen is with $$P_1=(M_1, 0) \\ P_2=(M_2,0) \\ P_3=\left(\sqrt{(m_3^2+p^2)},p\right) \\ P_4 =(|p|,-p)$$

where $P_1$ is the initial state and $P_2$, $P_3$, $P_4$ are the final. From here the example said

$$M_1 = M_2 + \sqrt{(m_3^2+p^2)} + |p|$$

Could somebody explain why we are allowed to say that, I thought that energy and momentum became "intertwined" and energy and momentum conversation were coupled into four momentum conservation.