So, I learn about rotation basic quaternion, and now I am trying to understand split-quaternion and their rotation. As far as I understand, there are different formula, like $q=N(cosh(a)+psinh(a))$. As far as I understand(again), I can ignore N if I take p as $(0;0;1)$ or similar. Does rotation of split-quaternion is the same as simple quaternion? Like, $qwq^-1$? And what about different types of split-quaternions?
If you can, please give some examples. I am bad at understanding without examples, just by basic formula for each situation... Thank you!
Here is an attempt to give some ideas on your question: It is well-known that rotations in Euclidean 3-space ($\mathbb{R}^3$) can be expressed by using unit-quaternions. A similar fact works for split-quaternions but in Lorentz-Minkowski 3-space ($\mathbb{R}^{1,2}$) which is defined as a real vector space with Lorentzian inner product: $$ \langle u,v\rangle_L=-u_1v_1+u_2v_2+u_3v_3\quad u=(u_1,u_2,u_3),v=(v_1,v_2,v_3). $$ In this space, the vectors are characterized by Lorentzian inner product. For a vector $v=(v_1,v_2,v_3)\in\mathbb{R}^{1,2}$, the vector $v$ is said to be a spacelike if $\langle v, v\rangle_L>0$ or $v=0$, timelike if $\langle v, v\rangle_L<0$, lightlike (or null) if $\langle v, v\rangle_L=0$.
I don't mention all definitions on split quaternions assuming that you already know. A split quaternion $q=q_0+q_1i+q_2j+q_3k$ can be expressed as $q=S_{q}+{\mathbf{V}_{q}},$ where $% S_{q}=q_{0}$ and $\mathbf{V}_{q}=q_{1}i+q_{2}j+q_{3}k$ are called the scalar part and vector part of $q$, respectively. Taking a split quaternion $q=q_0+q_1i+q_2j+q_3k$, we have that $I_q:=q\ast \bar{q}=\bar{q}\ast q=q_{0}^{2}+q_{1}^{2}-q_{2}^{2}-q_{3}^{2}$, where $\ast$ denotes the split-quaternion multiplication and $\bar{q}$ is the conjugate of $q$. A split quaternion $q$ is said to be spacelike, timelike or lightlike, if $I_{q}<0,I_{q}>0$ and $I_{q}=0,$ respectively. As you said, every split-quaternion has a polar form. For example, every timelike split quaternion with the spacelike vector part can be expressed as
\begin{equation*} q=N_q(\cosh \theta +\eta\sinh \theta), \end{equation*} where $\cosh\theta=\dfrac{q_0}{N_q}$, $\sinh\theta=\dfrac{\sqrt{-q_1^2+q_2^2+q_3^2}}{N_q}$, $\eta=\dfrac{q_1i+q_2j+q_3k}{\sqrt{-q_1^2+q_2^2+q_3^2}}$ is a unit spacelike vector in $\mathbb{R}^{1,2}$ and $\eta\ast \eta=1$.
After some basic facts, I would say that "Any rotation in $\mathbb{R}^{1,2}$ can be stated with the aid of a unit timelike split quaternion."
We denote the set of all timelike split quaternions by \begin{equation*} \mathbb{T}\mathbb{H}'=\{q=(q_0,q_1,q_2,q_3)\in\mathbb{H}'|q_0,q_1,q_2,q_3\in \mathbb{R}, I_q>0\}. \end{equation*} which forms a group under the split quaternion product. Consider the map $\phi$ for $q\in\mathbb{H}'$ unit split quaternion defined by \begin{equation*} \phi:\mathbb{R}^{1,2}\rightarrow\mathbb{R}^{1,2}; \quad \phi(\mathbf{x})=q\ast \mathbf{x} \ast q^{-1}, \quad \mathbf{x}\in\mathbb{R}^{1,2}. \end{equation*} Therefore, if $q=(q_0,q_1,q_2,q_3)$ is a unit timelike split quaternion, the corresponding rotation matrix can be expressed as \begin{equation} M_q =\left[ \begin{array}{ccc} q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2} & 2q_{3}q_{0}-2q_{1}q_{2} & -2q_{0}q_{2}-2q_{1}q_{3} \\ 2q_{1}q_{2}+2q_{3}q_{0} & q_{0}^{2}-q_{1}^{2}-q_{2}^{2}+q_{3}^{2} & -2q_{2}q_{3}-2q_{1}q_{0} \\ 2q_{1}q_{3}-2q_{1}q_{0} & 2q_{1}q_{0}-2q_{2}q_{3} & q_{0}^{2}-q_{1}^{2}+q_{2}^{2}-q_{3}^{2} \end{array} \right]. \end{equation}
I know it is a long answer! If you have still confusing let me know.