subscript $f$ is the property of the frame. We will give the frame an origin $\mathbf {r_f}$ so we can get the particle’s relative position in the rotating frame.
subscript $0$ is in inertial frame
no subscript is in the rotating frame
So we can write the relative position of the particle as $\mathbf{r}=\mathbf{r_0}-\mathbf{r_f}$
The inertial frame is related to the rotating frame:
<aside>
$$ \frac{d\mathbf{r}}{dt}=\boldsymbol{\omega}\times \mathbf{r}+\frac{\delta \mathbf{r}}{\delta t} $$
</aside>
$\delta$ is a derivative in the rotating frame
This only works for the relative position because we can ignore translation of the frame’s origin
Velocity:
$$ \renewcommand\vec\mathbf \frac{d\vec{r_0}}{dt}=\frac{d\vec{r_f}}{dt}+\frac{d\vec{r}}{dt}=\vec{v_f}+\vec{\omega}\times \vec{r}+\frac{\delta \vec{r}}{\delta t} $$
Differentiating,
$$ \renewcommand\vec\mathbf \newcommand\b\boldsymbol \begin{align*}\frac{d^2\vec{r_0}}{dt^2}&=\mathbf{a_f}+\frac{d\vec{\omega}}{dt} \times \vec r +\b\omega\times \frac{d\vec r}{dt}+\frac{d}{dt}\left(\frac{\delta \vec r}{\delta t}\right)\\&=\vec{a_f}+\b \alpha\times\vec r+\b\omega\times\left(\b\omega\times\vec r+\frac{\delta \vec r}{\delta t}\right)+\b\omega\times \frac{\delta \vec r}{\delta t}+\frac{\delta^2 \vec r}{\delta t^2}\\&=\vec{a_f}+\b \alpha\times\vec r+\b\omega\times(\b\omega\times\vec r)+2\b\omega\times \vec v+\vec a\end{align*} $$
Using Newton’s 2nd law, $\because \mathbf{F}=m\frac{d^2\mathbf{r_0}}{dt}$
<aside>
$$ \newcommand\b\boldsymbol \mathbf{a}=\mathbf F/m-\mathbf{a_f} -\b\alpha\times\mathbf{r}-\b\omega\times(\b\omega\times\mathbf{r})-2\b\omega\times \mathbf{v} $$
</aside>
The terms are, in order, external force, translational force, azimuthal force, centrifugal force, coriolis force.
Fictitious forces produce a net force as if all the mass was concentrated at the center of mass (as the cross products distribute over addition)
However, they can produce an extra net torque as the position vector appears in two locations in the torque equation (this is just a special form of the question of whether forces can be combined to produce the same net force and torque everywhere). Hence it is best to think about torque in an inertial frame to remove these issues.