Lie derivative of curvature

Question

Let $M$ be a Kähler manifold, with Kähler metric $g$. Let $X$ be a holomorphic Killing vector field of $g$, i.e. $\mathcal{L}_{X} g = 0$, where $\mathcal{L}_{X}$ is the Lie derivative along $X$. Let $R$ be the Riemannian curvature tensor of $g$. Is $\mathcal{L}_{X} R = 0$?