or via Prop. \[prop:existence\_of\_elliptic\_tori\_in\_special\_parahorics\], where these assumptions are of more techinical nature rather than of intrinsic nature (e.g. in their proofs, we verify certain statements by case-by-case analysis, where the possible cases are restricted or reduce substantially under such assumptions). It would be interesting to know if one can remove either of these conditions in these statements.

We remark that our sign convention up to Section 6 is the same as that of Langlands-Rapoport in [@LR87]; so for example, it is opposite to that of Kisin in [@Kisin17]. In Section 7, we also need to fix some more sign conventions, especially sign normalization of the local Langlands correspondence for tori (see Footnote \[ftn:LLC\_sign\] for our choice).

This article is organized as follows. The second section is a preliminary discussion, devoted to a review of some basic objects, including Kottwitz and Newton maps (defined for algebraic groups over $p$-adic fields), parahoric groups (in the Bruhat-Tits theory), extended affine Weyl groups, and $\{\mu\}$-admissible set. In the third section, we attempt to give a self-contained overview of the notions of Galois gerbs, the pseudo-motivic Galois gerb, admissible morphisms, Kottwitz triples, and admissible pairs, following closely the original source [@LR87]. At the same time, 