  Let $(G,X)$ be a Shimura datum of Hodge type, and $\mathscr{S}_K(G,X)$ its
integral model with hyperspecial (resp. parahoric, assuming the group is
unramified) level structure. We prove that $\mathscr{S}_K(G,X)$ admits a closed
embedding, which is compatible with moduli interpretations, into the integral
model $\mathscr{S}_{K'}(\mathrm{GSp},S^{\pm})$ for a Siegel modular variety. In
particular, the normalization step in the construction of $\mathscr{S}_K(G,X)$
is redundant. In particular, our results apply to the earlier integral models
constructed by Rapoport, Kottwitz etc. (resp. Rapoport-Zink etc.), as those
models agree with the Hodge type integral models for appropriately chosen
Shimura data.
  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.
