  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.
