Let $T_y(a)$ be a translation operator of a displacement $a$ parallel to the y-axis. In other words,
$$T_y(a)\vec{r}=\vec{r}+a\;\hat{y}$$
If $R_x(\theta)$ is a rotation of $\theta$ around the x-axis, how can I show that
$$R_x(\theta)T_y(a)R_x(-\theta)$$
is a translation along some axis? And how can I determine which axis is it?
[EDIT]
Thanks to mavzolej I was able to determine that the product of those operators generate a translation operator of the type
$$T_{e}(a)\vec{r}= \vec{r} + a\;\hat{e}$$
where $\hat{e}$ is the axis of translation defined by
$$\hat{e} = \cos\theta\;\hat{y} + \sin\theta\;\hat{z}$$
Knowing that, how can I use this result to deduce the commutation relation $[J_x,P_y]=iP_z$ ?
You can use the definition of J and P as the both are the infinitesimal generators for the both rotation and translation respectively.