Deriving interaction vertices in higher derivative theories

We derive cubic interaction vertices for a class of higher-derivative theories involving three arbitrary integer spin fields. This derivation uses the requirement of closure of the Poincar\`e algebra in four-dimensional flat spacetime. We find two varieties of permitted structures at the cubic level and eliminate one variety, which is proportional to the equations of motion, using suitable field redefinitions. We then consider soft theorems for field theories with higher-derivative interactions and construct amplitudes in these theories using the inverse-soft approach.