Abelian gauge-like groups of $L_\infty$-algebras

Given a finite type degree-wise nilpotent $L_\infty$-algebra, we construct an abelian group that acts on the set of Maurer-Cartan elements of the given $L_\infty$-algebra so that the quotient by this action becomes the moduli space of equivalence classes of Maurer-Cartan elements. Specializing this to degree-wise nilpotent dg Lie algebras, we find that the associated ordinary gauge group of the dg Lie algebra with the Baker-Campbell-Hausdorff multiplication might be substituted by the underlying additive group. This additive group acts on the Maurer-Cartan elements, and the quotient by this action yields the moduli space of gauge-equivalence classes of Maurer-Cartan elements.