In the paper "A Primer on the Differential Calculus of 3D orientations" by Bloesch et al. they give an example on how to apply the identities derived for the derivatives of various rotation expressions on some motion equations. Specifically for equation (48) we have
$$ \begin{align} {}_{B} \dot{\pmb{v}}_B &= \frac{d}{dt}\bigg(\phi_{IB}^{-1}\big({}_I \pmb{v}_B\big)\bigg) \\ &= \phi_{IB}^{-1}\big({}_I \pmb{\dot{v}}_B \big) - \big( {}_I \pmb{\dot{v}}_B \big)^{\times} C\big(\phi_{IB}\big)^{\top} {}_{I}\omega_{BI}\\ \end{align} $$
Where $\phi$ is an element of $SO(3)$, $C(\phi)$ is the 3x3 matrix representation of $\phi$, ${}_{I}\pmb{v}_B$ is the velocity of the body in the inertial frame.
It is explained that to do so we apply the product rule followed by the chain rule and various identities to get to the second line. However, I am a bit at a loss as to how the product rule applies when applying an element of $SO(3)$ to a vector.
Can someone show me all the steps to get to the second line?