I have a set $F$ of vector fields. The commutator $[v, u]$ is linear in $v$ and $u$ point by point, i.e., for each couple of vector fields $v\in F$ and $u\in F$, there are two scalars $a$ and $b$ such that:
$$
[v, u] = a v + b u
$$
I remark that, by "scalar", I mean that $a$ and $b$ are functions of the place, i.e. they depend on the position. They are not constant all over the manifold. Moreover, I remark that $a$ and $b$ can be different for each couple of vector fields.


If I'm not wrong, this is not a general property of vector fields. Does it have a name? Are there known properties of such a set?


I can add that $F$ contains as many elements as the dimension of the space; moreover, they are linearly independent point by point. This means that any vector field $u$ can be written as the sum of the $v\_i\in F$
multiplied by suitable scalars (functions of place) $a\_i$. Are there further properties in this case?



Set of vector fields with a particular expression of the commutator
Extended comment as asked by OP.
Let's call your manifold $X$, $\dim X = n$ .
Your assumption is that there are $v\_1,\dots, v\_n\in \mathcal{T}(X)$ (vector fields) that are pointwise linearly independent and moreover
$[v\_i,v\_j] = a\_{i,j} v\_i + b\_{i,j} v\_j$ for $a\_{i,j}, b\_{i,j} \in C^{\infty}(X)$.


*Remark 1.* since $v\_1,\dots, v\_n$ are linearly independent $X$ is parallelizable. It means that its tangent bundle $TX \simeq X\times \mathbb{R}^n$. This is a global property and is not true in general, for example the even-dimensional spheres $\mathbb{S}^{2k}$ do not have this property because you cannot even find a single never vanishing vector field over $\mathbb{S}^{2k}$.


Now, consider a pair $v\_1,v\_2$. At each $x\in X$, consider $D\_x = Span(v\_1(x), v\_2(x))$ this is a 2-dimensional subspace of the tangent space $T\_x X$ which varies smoothly within $x$, in other words a *distribution*.


The condition $[v\_1,v\_2] = a\_{1,2} v\_1 + b\_{1,2} v\_2$ now says that $[v\_1,v\_2](x) \in D\_x$. Therefore Frobenius theorem applies telling us that $D$ is an integrable distribution.


Being integrable, it means in particular that $X = \bigsqcup\_\alpha S\_\alpha$ where $S\_\alpha$ are disjoint 2-dimensional maximally integrable (immersed) submanifolds for $D$.
In other words, for each $x\in X$ there is an $S\_\alpha$ s.t. $T\_x S\_\alpha = D\_x =Span (v\_1,v\_2)$, the tangent space coincides with the distribution.


Notice that I am considering *maximal* integrable submanifolds, this is the same thing as when you solve an ODE and you look for maximal solutions.
For the definition of immersed submanifold see my comment below. Keep in mind that $S\_\alpha $ is image of a 2-manifold $N\_\alpha$ under an immersion map $f\_\alpha$. $S\_\alpha$ may not be a manifold with the topology inherited by $X$ hence calling them surfaces is an abuse of language. When I say $T S\_\alpha$ I mean $d f\_\alpha (T N)$ (think of $f\_\alpha$ as a parametrization).


The surfaces $S\_\alpha$ are invariant under the flow of $v\_1,v\_2$ since they are now tangent vector fields to $S\_\alpha$ hence you can solve the flow equations over $S\_\alpha$.


We can also say something about the topology of the surfaces $S\_\alpha$ (to be precise about the topology of $N\_\alpha$).
Indeed since $v\_1,v\_2$ are linearly independent, their restriction to $S\_\alpha$ provides a global framing for $S\_\alpha$ since $\forall s \in S\_\alpha, Span(v\_1(s), v\_2(s)) = T\_s S\_\alpha$ and $v\_1,v\_2$ are globally defined over $S\_\alpha$. This shows that $S\_\alpha$ (read $N\_\alpha$) is parallelizable.


The only closed, parallelizable 2-manifolds are tori $\mathbb{S}^1\times \mathbb{S}^1$ hence if one of the $S\_\alpha$ is closed, it must be a torus.
Note that the topology of the $S\_\alpha$ may change with $\alpha$.


If $S\_\alpha$ is not closed, then parallelizability tells us that $S\_\alpha$ must be orientable (hence no Mobius bands). Also all characteristic classes of $T N\_\alpha$ must vanish but in dimension two for non-closed surfaces this doesn't say anything more than orientability (for higher dimensions you get more constraints).


Final note. Consider now $v\_1,\dots, v\_k$, $k<n$. They induce a distribution of $k$-planes, $D\_x = Span (v\_1,\dots, v\_k)$.
As above this distribution is involutive: $[\xi,\zeta](x) \in D\_x $ for any $\xi,\zeta$ sections of $D$. Hence we can apply Frobenius and find that $X$ is foliated $X = \bigsqcup\_\alpha S\_\alpha$ with $\dim S\_\alpha=k$. The $k$-manifolds $N\_\alpha$ will be parallelizable as above and invariant under the flow of $v\_1,\dots, v\_k$.


