Shimura lift of Rankin–Cohen brackets of eigenforms and theta series

The Shimura lift of a Hecke eigenform multiplied by a theta series is the square of the form. We extend this result by generalizing the product map to the Rankin–Cohen bracket. We prove that the Shimura lift of the Rankin–Cohen bracket of an eigenform and a theta series is given by the Rankin–Cohen bracket of the eigenform and itself.