Internal symmetry of the L₃ algebra arising from a Lie pair

A Lie pair is an inclusion A to L of Lie algebroids over the same base manifold. In an earlier work, the third author with Bandiera, Stienon, and Xu introduced a canonical L-<= 3 algebra Gamma(Lambda(center dot)A(boolean OR) circle times L/A) whose unary bracket is the Chevalley-Eilenb erg differential arising from every Lie pair (L, A). In this note, we prove that to such a Lie pair there is an associated Lie algebra action by Der(L) on the L-<= 3 algebra Gamma(boolean AND(center dot)A(boolean OR) circle times L/A). Here Der(L) is the space of derivations on the Lie algebroid L, or infinitesimal au-tomorphisms of L. The said action gives rise to a larger scope of gauge equivalences of Maurer-Cartan elements in Gamma(boolean AND(center dot)A boolean OR circle times L/A), and for this reason we elect to call the Der(L)-action internal symmetry of Gamma(boolean AND(center dot)A(boolean OR) circle times L/A).