Some automorphism groups are linear algebraic

Consider a normal projective variety $X$, a linear algebraic subgroup $G$ of $\mathrm{Aut}(X)$, and the field $K$ of $G$-invariant rational functions on $X$. We show that the subgroup of $\mathrm{Aut}(X)$ that fixes $K$ pointwise is linear algebraic. If $K$ has transcendence degree $1$ over the base field $k$, then $\mathrm{Aut}(X)$ is an algebraic group.