  Let $(M,g)$ be a compact K\"ahler manifold and $f$ a positive smooth function
such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the
K\"ahler form $\omega_g$ is a holomorphic Killing vector field. We say that the
pair $(g,f)$ is conformally Einstein-Maxwell K\"ahler metric if the conformal
metric $\tilde g = f^{-2}g$ has constant scalar curvature. In this paper we
prove a reductiveness result of the reduced Lie algebra of holomorphic vector
fields for conformally Einstein-Maxwell K\"ahler manifolds, extending the
Lichnerowicz-Matsushima Theorem for constant scalar curvature K\"ahler
manifolds. More generally we consider extensions of Calabi functional and
extremal K\"ahler metrics, and prove an extension of Calabi's theorem on the
structure of the Lie algebra of holomorphic vector fields for extremal K\"ahler
manifolds. The proof uses a Hessian formula for the Calabi functional under the
set up of Donaldson-Fujiki picture.
