On the extendability of locally defined isometries of a pseudo-Riemannian manifold

Let $\eta$ be a stationary subalgebra of the Lie algebra $\zeta$ of all Killing vector fields on a pseudo-Riemannian analytic manifold, $G$ be a simply connected Lie group generated by the algebra $\zeta $, and $H$ be its subgroup generated by the subalgebra $\eta$. Then the subgroup $H$ is closed in $G$.