Factorization of cubic vertices involving three different higher spin fields

We derive a class of cubic interaction vertices for three higher spin fields, with integer spins $\lambda_1$, $\lambda_2$, $\lambda_3$, by closing commutators of the Poincar\'e algebra in four-dimensional flat spacetime. We find that these vertices exhibit an interesting factorization property which allows us to identify off-shell perturbative relations between them.