In the paper "Homotopy Lie algebras", Schechtman and Hinich proved that any cosimplicial differential graded Lie algebra has the structure of a 'Lie May algebra'.
If my understanding is right here, a Lie May algebra is just an uncommon term for Lie infinity algebra.
Now the question is, if there is any explicit construction of this Lie infinity algebra structure?
I mean, are there any explicit formulas that define the $k$-ary brackets of this Lie $\infty$-algebra in terms of the cofaces/codegeneracies and the Lie bracket of the underlying cosimplicial DG Lie algebra?
Edit: If this is unknown, then I would be thankful for ideas on how to proceed in an attempt to find these maps.