is possible due to the “geometrical” results of [@Kisin17] which are often quite strong (“geometrical” in the sense that it can be formulated without using the language of Galois gerbs). Here, we mention one such result, [@Kisin17 Cor.2.2.5] which we have dubbed “strong CM lifting theorem” to distinguish it from the usual CM lifting theorem [@Kisin17 Thm.2.2.3] (which only says that every isogeny class ${\mathscr{I}}$ in a SV of Hodge type over ${\bar{\mathbb{F}}_{p}}$ contains a point which lifts to a CM point).

Generalization of results of [@LR87]
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The main results of [@LR87] assume that the level subgroup ${\mathbf{K}}_p$ is hyperspecial (so, in particular that $G_{{{\mathbb Q}_p}}$ is unramified) and that the derived group is simply connected. To extend the scope of our method of proof beyond these cases, our first primary task in this article is to generalize their works to more general parahoric levels (so as to allow possibly bad reductions) under the assumption that $G_{{{\mathbb Q}_p}}$ is quasi-split and spits over a tamely ramified extension of ${{\mathbb Q}_p}$ (in fact, we will assume less than that, see Thm. \[thm:LR-Satz5.3\] for a precise condition), and also to remove the restriction on the derived group; some of our results 