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.