Brace $B_{\infty}$ algebras associated with Hopf algebroids

We apply the operadic modeling of brace $B_{\infty}$ algebras, as developed by Gerstenhaber and Voronov, to the context of Hopf algebroids in the sense of Xu. Specifically, we construct a strict $B_{\infty}$ isomorphism between the type I and type II twisted brace $B_{\infty}$ algebras arising from any twistor of a Hopf algebroid. As an application of this framework, we examine two specific brace $B_{\infty}$ algebras derived from Lie algebra pairs, and reveal previously obscured relationships between them. One of these is the dg Lie algebra governing deformations of algebraic dynamical twists, while the other arises from the quantum groupoid comprised of particular invariant differential operators.