 Solution

The three on-shell spatial momenta have nine components. Four conservation constraints leave five, and an unobserved overall spatial rotation removes three for the scalar decay, leaving two internal kinematic variables. The pair invariants obeyThus two determine the third. They also determine each rest-frame daughter energy,and hence all momentum magnitudes and pairwise angles. The momentum triangle is fixed up to overall orientation. The Dalitz plot describes this internal kinematics, rather than claiming to record three independently oriented Four-vectors. Polarized parents or additional measured spin information can require orientation variables beyond the two invariant axes.