The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan formalism indeed produces a deformation problem and conversely for most deformation problems the dg-Lie algebra is well known. In the derived setting this principle has even become a theorem following Lurie-Prydham. However, I was wondering if anyone could explain the original intuition Deligne had behind this principle?

The deformation theoretic principle that any reasonable deformation problem should be governed by a dg-Lie algebra seems to come from a letter of Deligne to Millson. It is clear how the Maurer-Cartan formalism indeed produces a deformation problem and conversely for most deformation problems the dg-Lie algebra is well known. In the derived setting this principle has even become a theorem following Lurie-Pridham. However, I was wondering if anyone could explain the original intuition Deligne had behind this principle?