# Lie algebra preserving ideal of functions

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.