About the puzzle---I think selfAdjoint has already figured it out but I did not follow all the discussion, so I will take a guess.
Someone, I think it was JB, suggested using letters T and S for "twist" and "shift", where T is an element of the Lorentz group and S is thought of as a translation (we are building the Poincaré group)
[EDIT] GROAN. I just looked at the "Higher Yang Mills" paper that I lost a couple of days ago, while cleaning up the living room. the Poincaré 2-group is explained as EXAMPLE 9. So the puzzle was already answered.
...I talked to Kea and she suggested one by Girelli and Pfeiffer, which I never got around to looking at it. Maybe I will now.
Higher gauge theory -- differential versus integral formulation
Florian Girelli, Hendryk Pfeiffer
J.Math.Phys. 45 (2004) 3949-3971
"The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses non-Abelian 1- and 2-forms in order to generalize the connection 1-form of a conventional gauge theory to the next level. The integral picture makes use of curves and surfaces labeled with elements of non-Abelian groups and generalizes the formulation of gauge theory in terms of parallel transports..."
I hope Kea is better now and was wishing she would suddenly materialize amidst this thread.
Well, looking at Girelli/Pfeiffer, I see right away references
 J. C. Baez: Higher Yang–Mills theory (2002). Preprint hep-th/0206130.
 J. C. Baez and A. Crans: Higher dimensional algebra VI: Lie 2-algebras (2003). Preprint math.QA/0307263.
My posts are just not helpful in this thread at this point. I shall delete the next one.