We can think hyperbolic 5-space as, $$\mathcal{H}^5=SO^+_{5,1}(\mathbb{R})/SO_5(\mathbb{R})=SL_2(\mathbb{H})/Sp^*_2(\mathbb{H}),$$$\mathbb{H}$ is real quaternion algebra. By Iwasawa Decomposition the orientation preserving isometry group of $\mathcal{H}^5$ is $$G=PSL_2(\mathbb{H})=NAK,$$where $$N=\left\lbrace n(x):=\begin{pmatrix}1&x\\&1\end{pmatrix}|x\in\mathbb{H}\right\rbrace\ \ \text{and}\ \ A=\left\lbrace a(y):=\begin{pmatrix}\sqrt y\\&\sqrt y^{-1}\end{pmatrix}|y\in\mathbb{R}^+\right\rbrace.$$

The quotient $G/K$, which could be realized as $$\left\lbrace \begin{pmatrix} y&x\\&1\end{pmatrix}|y\in\mathbb{R}^+,x\in\mathbb{H}\right\rbrace$$gives a realization of $\mathcal{H}^5$. Define $I(P)=y$ for $P\simeq n(x)a(y)$

Many literatures use that $G$ acts on $\mathcal{H}^5$ by Mobius Transformation, i.e. $$gP=\begin{pmatrix} a&b\\c&d\end{pmatrix}P=(aP+b)(cP+d)^{-1}.$$ My questions are,

1) What are the multiplication $(aP)$ or addition $(aP+b)$ rules being used here?

2) Can we think $P=x+*y$ (for some $*$) as we do in $\mathcal{H}^2 \ (*=i)$ and $\mathcal{H}^3 \ (*=j)$?

3) Is it true that $\frac{I(gP)}{I(P)}=\frac{||\det(g)||}{||cx+d||^2+||cy||^2}$ where ||.|| is quaternionic norm? (This happens in $\mathcal{H}^2$ and $\mathcal{H}^3$).

Thanks.