For $(X,g)$ a Riemannian manifold or pseudo-Riemannian manifold its isometry group
is that subgroup of the group of all diffeomorphisms $\phi : X \to X$ that are isometries: which preserve the metric $g$ in that
The Lie algebra of $Iso(X,g)$ is spanned by the Killing vectors of $(X,g)$.
The isometry group of Minkowski spacetime is the Poincaré group.
The isometry group of anti de Sitter spacetime is the anti de Sitter group.
The isometry group of de Sitter spacetime is the de Sitter group.