Lie algebra of killing vector fields and its stationary subalgebra

Let $\mathfrak{g}$ be the Lie algebra of all Killing vector fields on a locally homogeneous, analytic Riemannian manifold $M$, $\mathfrak{h}$ be a stationary subalgebra of $\mathfrak{g}$, $G$ be the simply connected group generated by the algebra $\mathfrak{g}$, $H$ be the subgroup of $G$ generated by the subalgebra $\mathfrak{h}$, $\mathfrak{z}$ be the center of the algebra $\mathfrak{g}$, $\mathfrak{r}$ be its radical, and $[\mathfrak{g};\mathfrak{g}]$ be its commutator subgroup. If $\dim\big(\mathfrak{h}\cap\big(\mathfrak{z} + [\mathfrak{g}, \mathfrak{g}] \big)\big) = \dim \big(\mathfrak{h} \cap [\mathfrak{g}, \mathfrak{g}]\big)$, then $H$ is closed in $G$. If for any semisimple subalgebra $\mathfrak{p}\subset\mathfrak{g}$ satisfying the condition $\mathfrak{p}+\mathfrak{r}=\mathfrak{g}$, the relation $(\mathfrak{p}+\mathfrak{z})\cap\mathfrak{h} =\mathfrak{p}\cap\mathfrak{h}$ holds, then $H$ is closed in $G$. We also examine the analytic continuation of the given local, analytic Riemannian manifold.