Let (G, X) be a Shimura datum of Hodge type, and \(\mathscr {S}_K(G,X)\) its integral model with hyperspecial level structure. We prove that \(\mathscr {S}_K(G,X)\) admits a closed embedding, which is compatible with moduli interpretations, into the integral model \(\mathscr {S}_{K'}(\operatorname {GSp},S^{\pm })\) for a Siegel modular variety. More precisely, the normalization step in the construction of \(\mathscr {S}_K(G,X)\) is redundant, and the flat closure model is already smooth at hyperspecial level. As a consequence, this also removes the normalization step in the construction of \(\mathscr {S}_{K'}(G',X')\) when \((G',X')\) is of arbitrary abelian type. Moreover, combined with a result of Lan’s on the boundary components of toroidal compactifications of integral models, our result also implies that there exist closed embeddings of toroidal compactifications of integral models of Hodge type into toroidal compactifications of Siegel integral models, for suitable choices of cone decompositions.
