Isometry of a semi-Riemannian manifold

Let $(𝑀, 𝑔)$ be a [[Semi-Riemannian manifold|semi-Riemannian $𝐶 𝛼$ -manifold]]. An isometry $𝜑 \in I s o (𝑀, 𝑔)$ is a $𝐶 𝛼$ -diffeomorphism $𝜑 : 𝑀 \to 𝑀$ which preserves the metric, diff i.e. the pullback $𝜑 * 𝑔 𝑎 𝑏$ satisfyies $𝜑 * 𝑔 𝑎 𝑏 = 𝑔 𝑎 𝑏$ . The isometries form a Lie group $I s o (𝑀, 𝑔)$ , whose corresponding Lie algebra $𝔦 𝔰 𝔬 (𝑀, 𝑔)$ consists of Killing fields.