Normalization in integral models of Shimura varieties of Hodge type

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 $mathscr{S}_{K}(mathrm{GSp},S^{pm})$ 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 and Kottwitz, as those models agree with the Hodge type integral models for appropriately chosen Shimura data.