Algebraic properties of bounded killing vector fields

In this paper, we consider a connected Riemannian manifold $M$ where Lie group $G$ acts effectively and isometrically. Assume $X\in\mathfrak{g}=\mathrm{Lie}(G)$ defines bounded Killing vector field, find some crucial algebraic properties of the decomposition $X=X_r+X_s$ according to Levi $\mathfrak{g}=\mathfrak{r}(\mathfrak{g})+\mathfrak{s}$, $\mathfrak{r}(\mathfrak{g})$ is radical, $\mathfrak{s}=\mathfrak{s}_c\oplus\mathfrak{s}_{nc}$ subalgebra. The coincides with abstract Jordan $X$, unique in sense that it does not depend on choice $\mathfrak{s}$. By these properties, prove eigenvalues $\mathrm{ad}(X):\mathfrak{g}\rightarrow\mathfrak{g}$ are all imaginary. Furthermore, when $M=G/H$ homogeneous space, can completely determine fields induced by vectors $\mathfrak{g}$. We space fields, or equivalently $\mathfrak{g}$ for $G/H$, compact subalgebra, such its semi-simple part ideal $\mathfrak{c}_{\mathfrak{s}_c}(\mathfrak{r}(\mathfrak{g}))$ $\mathfrak{g}$, Abelian sum $\mathfrak{c}_{\mathfrak{c}(\mathfrak{r}(\mathfrak{g}))} (\mathfrak{s}_{nc})$ two-dimensional irreducible $\mathrm{ad}(\mathfrak{r}(\mathfrak{g}))$-representations $\mathfrak{c}_{\mathfrak{c}(\mathfrak{n})}(\mathfrak{s}_{nc})$ corresponding nonzero imaginary weights, i.e. $\mathbb{R}$-linear functionals $\lambda:\mathfrak{r}(\mathfrak{g})\rightarrow \mathfrak{r}(\mathfrak{g})/\mathfrak{n}(\mathfrak{g}) \rightarrow\mathbb{R}\sqrt{-1}$, $\mathfrak{n}(\mathfrak{g})$ nilradical.