  In this paper we prove a family of identities for Bernoulli numbers
parameterized by triples of integers $(a,b,c)$ with $a+b+c=n-1$, $n\ge 4$.
These identities are deduced while translating into homotopical terms the gauge
action on the Maurer Cartan Set which can be seen an abstraction of the
behaviour of gauge infinitesimal transformations in classical gauge theory. We
show that Euler and Miki's identities, well known and apparently non related
formulas, are linear combinations of our family and they satisfy a particular
symmetry relation.
