I'm slightly puzzled about the space in which the Lie derivatives of vector fields live. I get the worrying impression that it lives in the double tangent bundle, which I know is not true, so i need some clarification :)
Here is the reason:
Setting: Take $M$ a smooth manifold, $\pi:TM\rightarrow M$ its tangent bundle and $v,w:M\rightarrow TM$ smooth sections on that bundle, i.e. (tangent) vector fields. There exists for each $x\in M$ at least for small enough $t\in(a,b)$ with $a<0<b$ an integral curve for $v$. Let's call it $\phi_t(x)$. The Lie derivative is defined by: \begin{equation} (\mathcal L_vw)_x=\frac{d}{dt}\Big((\phi_{-t})_*w_{\phi_t(x)}\Big)_{t=0}\qquad = \frac{d}{dt}\big(\gamma_t\big)_{t=0} \end{equation} First observation: $\gamma_t$ is a curve in $T_x M$, because the differential $(\phi_{-t})_*$ acts on a vector $w_{\phi_t(x)}$ defined at $T_{\phi_{t}(x)}M$ and pushes it to a vector in $T_xM$. Hence $\gamma_t$ is $T_x M$-valued.
Second observation: the derivative of $\gamma_t$ lies in the tangent space of $T_xM$, which is $TT_xM$. So $\mathcal L_vw:X\rightarrow TTX$ is a section on the double tangent bundle. I find that the argument runs much in the same way as for tangent vectors of curves: $\phi_{(\cdot)}(x):\mathbb R\rightarrow M$ is a curve on $M$ and \begin{equation} "\frac{d}{dt}\phi_t(x)\Big|_{t=0}"\quad=(\phi_{(\cdot)}(x))_*\frac{d}{dt}\Big|_{t=0}=v(x) \end{equation} is the tangent vector of the curve, i.e. a pushforward of the canonical time derivative vector field to $TM$ along the curve.
So, where did I go wrong? In John Lee's "an introduction to Smooth manifolds" he seems to get around this issue, but only by arguing with canonical forms of regular vector fields, which I don't understand.
thanks! in anticipation of enlightment.