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Assume $X\\in\\mathfrak{g}=\\mathrm{Lie}(G)$ defines a bounded Killing vector field, we find some crucial algebraic properties of the decomposition $X=X_r+X_s$ according to a Levi decomposition $\\mathfrak{g}=\\mathfrak{r}(\\mathfrak{g})+\\mathfrak{s}$, where $\\mathfrak{r}(\\mathfrak{g})$ is the radical, and $\\mathfrak{s}=\\mathfrak{s}_c\\oplus\\mathfrak{s}_{nc}$ is a Levi subalgebra. 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