classical mechanics - Derivation of the time-derivative in a rotating frame of refrence

I have so trouble following Goldsteins derivation of the time derivative in the rotating refrence frame, and its use to derive the coriolis force (sec. 4.9-10) Given an intertial frame of refrence, $S$, and a rotating one, $S'$, contected through the transformation

$$ x_i' = a_{ij}(t)x_j, \\ x_i = a_{ij}'(t)x_j' = a_{ji} (t)x_j', $$

we can get the time derivative in the rotating refrence-frame by use of the chain rule,

$$ \frac{d}{dt}x_i = a_{ij}\frac{dx_j'}{dt}+ \frac{da_{ij}}{dt} x_j'. $$

Assuming $a_{ij}(0) = \delta_{ij}$, w can rewrite the equation as

$$ \frac{d}{dt}x_i = \frac{dx_i'}{dt} + \frac{da_{ij}}{dt} x_j. $$

$\mathrm{d} a_{ij}$ is an infinitesimal rotation, and can be writen as (Goldstein in sec. 4.8)

$$ \mathrm{d}A = \begin{bmatrix} 0 & \mathrm{d}\Omega_3 & -\mathrm{d}\Omega_2\\ -\mathrm{d}\Omega_3 & 0 & \mathrm{d}\Omega_1 \\ \mathrm{d}\Omega_2 & -\mathrm{d}\Omega_1 & 0 \end{bmatrix} = \pmb{\omega} dt, $$

so that $\mathrm{d}a_{ij} = \epsilon_{ikj} \omega_k \mathrm{d}t$. This finally gives us

$$ \frac{d}{dt}x_i = \frac{dx_i'}{dt} + \epsilon_{ikj} \omega_k x_{j} = \bigg(\frac{d}{dt}a_{ji} + \epsilon_{ikj} \omega_k \bigg)x_{j}, $$

or as Goldstein writes it,

$$ \bigg(\frac{d}{dt}\bigg)_{space} = \bigg(\frac{d}{dt}\bigg)_{body} + \omega \times. $$

He then quickly uses this operator form to get

$$ \frac{d}{dt} \pmb{r} = \pmb v = \bigg(\frac{d}{dt} \pmb{r} \bigg)_{body} + \pmb\omega \times \pmb r = \pmb v' + \pmb \omega \times\pmb r', $$

(notice the switch between unprimed and primed $\pmb r$) and then

$$ \frac{d}{dt} \pmb v = \bigg(\frac{d}{dt} \pmb v \bigg)_{body} + \pmb \omega \times \pmb v = \bigg(\frac{d}{dt} (\pmb v' + \pmb \omega \times\pmb r') \bigg)_{body} + \pmb \omega \times (\pmb v' + \pmb \omega \times\pmb r') \\ = \pmb a' + 2 \pmb \omega \times \pmb v' + \pmb \omega \times(\pmb \omega \times \pmb r'). $$

I have tried to derive this using the indecies, but to no avail. If anyone is able to this, it would be greatly apriciated. I am espeacially having trouble with the term

$$ \bigg(\frac{d}{dt} (\pmb v' + \pmb \omega \times\pmb r') \bigg)_{body}. $$

I also find Goldsteins switch between the primed and unprimed system dubious, so any clarafications on this would also help.