On infinitesimal automorphisms of almost symplectic structures

On the tangent bundle $TM$ of a manifold $M$ endowed with an almost symplectic structure $\omega$ and a linear connection $\nabla$ compatible with $\omega$, we consider the Riemannian metric $G$ which is Hermitian with respect to the canonical almost complex structure $J$ and the corresponding almost symplectic structure $\Omega$. We study the infinitesimal automorphisms of these structures on $TM$, and, in particular, prove that the dimension of the Lie algebra of natural automorphisms of $G$ and of $\Omega$ is less than or equal to $n(n+3)/2$.