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.