I am studying some material about rotating frames and I need to compute the following quantity: $$\frac{d}{dt}R(t)=\frac{d}{dt}e^{\vec{\theta}(t)\cdot \vec{J}}$$ where $\vec{J}$ is the 3-tuple of the $3\times 3$ generators of rotation. To be able to use the simple rule of exponential derivatives, I need to show $[\vec{\theta}\cdot \vec{J},\vec{\omega}\cdot \vec{J}]=\vec{J}\times (\vec{\theta}\times \vec{\omega})=0$, which would be true if $\vec{\theta}\times \vec{\omega}$ vanishes. However, while it seems intuitively true, I am not sure why $\vec{\theta}$ and $\vec{\omega}$ ought to be parallel when the axis of rotation is changing.
