Condition for composition of vector fields to be a vector field

Question

I'm studying the Lie bracket and was wondering if there's some condition on the space so that the composition of vector fields is again always a vector field. This is what I have so far:

Let $X=\sum\limits_{i} a^i \partial_i$ and $Y=\sum\limits_{j} b^j \partial_j$ be two vector fields. After some algebra we get

$$ (XY)f = \Big( \sum\limits_{i,j} \Big( (a^j \partial_j b^i)\partial_i + (a^jb^i)\partial_{ij} \Big) \Big)f $$

So $XY$ is a vector field if we can write

$$ \partial_{ij} = \sum\limits_{k} c^k \partial_k, \quad \text{for some} \ c^k\colon \mathbb{R} \to \mathbb{R}. $$

i.e, if we can write

$$ \partial_{ij} f - \sum\limits_{k} c^k \partial_k f =0, \quad \forall f.$$

I'm just wondering if there's a way to imagine such a space or if there's a simpler way to explore what I'm trying to do.

Answer 1

This cannot work, since you can always take a function $f$ such that $f$ and $df$ vanish in a point, while all second derivatives in that point are non-zero.