Inertial forces
$$\vec{r}=\vec{r}'+\vec{r}_0,\quad\quad r^i\vec{e}_i={r'}^i\vec{e}'_i+r_0^i\vec{e}_i,\quad\quad\frac{d}{dt}\vec{e'}_i=\vec{\omega}\times\vec{e'}_i$$
$$\frac{dr^i}{dt}\vec{e}_i=\frac{d{r'}^i}{dt}\vec{e}'_i+\vec{\omega}\times\vec{r}'+\frac{dr_0^i}{dt}\vec{e}_i,\quad\quad\vec{v}=\vec{v}'+\vec{\omega}\times\vec{r}'+\vec{v}_0$$
$$\frac{d^2r^i}{dt^2}\vec{e}_i=\frac{d^2{r'}^i}{dt^2}\vec{e}'_i+\vec{\omega}\times\vec{v}'+\frac{d\vec{\omega}}{dt}\times\vec{r}'+\vec{\omega}\times\vec{v}'+\vec{\omega}\times\left(\vec{\omega}\times\vec{r}'\right)+\frac{d^2r_0^i}{dt^2}\vec{e}_i$$
$$\vec{a}=\vec{a}'+2\vec{\omega}\times\vec{v}'+\frac{d\vec{\omega}}{dt}\times\vec{r}'+\vec{\omega}\times\left(\vec{\omega}\times\vec{r}'\right)+\vec{a}_0$$