Does the Nagano-Sussmann orbit theorem--typically stated for real manifolds, as it is used in optimal control--continue to hold for complex manifolds?

Question

The Nagano-Sussmann theorem states that, given a point $x_{0}$ on a real manifold $M$, and a Lie algebra $L$ generated by a family ${\cal F}$ of smooth vector fields on $M$, the orbit ${\cal O}_{x_{0}}$ of $x_{0}$ under the action of $L$ is an immersed submanifold of $M$. It would seem that the answer is Yes, as the proof for the real case seems to admit a complex counterpart in a straightforward fashion, at least for vector fields smooth enough over the complex field. But, I just wanted to double-check: might I be missing something?