Let $G$ be an algebraic group acting on an affine variety $X=\operatorname{Spec}A$ (all over $\mathbb{C}$). This gives an action of $G$ on the $\mathbb{C}$-algebra $A$, and an action of the Lie algebra $\mathfrak{g}$ of $G$ on $A$ by derivations.

If the action is not transitive, then $G$ will preserve some nontrivial ideal of $A$ (namely take an ideal of a closed $G$-orbit). In particular, $\mathfrak{g}$ will preserve this ideal.

My question is whether this remains true if we only have the lie algebra and no group action. In particular, suppose that $\mathfrak{g}$ is a finite-dimensional Lie subalgebra of $\operatorname{Der}_{\mathbb{C}}(A)$, and suppose that it does not act 'transitively' on $X$, i.e. for some closed point $x\in X(\mathbb{C})$ the natural map $$ \mathfrak{g}\to T_xX $$ is not surjective, where $T_xX$ is the tangent space of $X$ at $x$. Then, must there exist a non-trivial ideal $I$ of $A$ which is preserved by $\mathfrak{g}$? Feel free to assume $X$ is smooth, say. Note I am not assuming the action of $\mathfrak{g}$ on $A$ is integrable, else we could use the statement about group actions stated initially.