we generalize these notions and establish their properties (notably, admissible morphism and Kottwitz triple) beyond the original assumption that the group has simply connected derived group. We also give a statement of the Langlands-Rapoport conjecture, as formulated by Rapoport [@Rapoport05 $\S$8] so as to cover parahoric levels. Along the way, we extend results on special admissible morphisms to (special maximal) parahoric levels, under the assumption that $G_{{{\mathbb Q}_p}}$ is quasi-split. In Section 4, we prove Theorem \[thm\_intro:1st\_Main\_thm\], (1) above, namely that every admissible morphism is conjugate to a special admissible morphism (in our case of general parahoric level), as well as the fact that every admissible pair is nested in a special Shimura datum. For some other potential applications in mind, we spilt the proof into a few steps and formalize each of them into a separate proposition (incorporating slight improvements). The results in this section are in large part translations of the original results, except for generalizations to our setting and reorganization (with small improvements). However, some of the generalizations, e.g. the proof of Lemma \[lem:LR-Lemma5.11\], are rather non-trivial. In Section 5, we establish the first version of effectivity criterion of Kottwitz triple (Thm. \[thm:LR-Satz5.25\]). In Section 6, we 