I have two (or more) smooth and integrable vector fields $v,w$ on a smooth manifold $M$. Each generates a flow map $\Phi_v$,$\Phi_w$ that forms a single parameter Lie group of diffeomorphisms. Let's call these groups $G_v$ and $G_w$
If we assume that $v$ and $w$ commute ($[v,w]=0$) and are linearly independent as tangent vectors on all points of $M$, then the collection of all flows generated by linear combinations of $v$ and $w$ should also be a Lie group of diffeomorphisms, this time of dimension two.
My question is, can this compound Lie group $G_{v,w}$ be constructed just from the Lie groups $G_v$ and $G_w$? I didn't find any way to do that using standard products, but I might be looking at this the wrong way.