Find the matrices of irreducible quaternionic representations of $SU(2)$ The irreducible complex representation of the group $\mathrm{SU}(2)$ of dimension $2n$, $n\geq 1$, is of quaternionic type. In particular, it can be represented by quaternionic matrices $W(g)$, $g \in\mathrm{SU}(2)$. How to calculate their matrix entries?
I know the answer for $n=1$. Let $g\in\mathrm{SU}(2)$, and let $q\in\mathrm{Sp}(1)\subset\mathbb{H}$ be the image of $g$ under the standard isomorphism $\mathrm{SU}(2)\to\mathrm{Sp}(1)$. Then $W(g)=q$.
When $n\geq 2$, the answer must be a $n\times n$ matrix $W$ with quaternionic entries, where each entry $W_{ij}$ is a function of the quaternionic variable $q\in\mathrm{Sp}(1)$. How to calculate $W_{ij}(q)$?   
 A: The plan of the answer.


*

*By Theorem 3.57 from J. Adams, Lecture on Lie groups, the complexification $c'W$ is irreducible. Under complexification, the matrix entry $W_{mn}(g):=a(g)+b(g)\mathrm{j}$ becomes the matrix $\left(\begin{smallmatrix}a(g) & -\overline{b(g)} \\ b(g) & \overline{a(g)}\end{smallmatrix}\right)$ with $a$, $b\in\mathbb{C}$. We identify
$$
a(g)=D^{\ell}_{mn}(g),\qquad b(g)=D^{\ell}_{-m\,n}(g),
$$
where $\ell=n-1/2$, ant the right hand sides are the Wigner $D$-functions, that is, the matrix elements of irreducible unitary representations of $SU(2)$. Conversely, 
$$
W_{mn}(g)=D^{\ell}_{mn}(g)+D^{\ell}_{-m\,n}(g)\mathrm{j},\qquad m,n=1/2,3/2,\dots,\ell.
$$

*Usually, the Wigner $D$-functions are given in terms of the Euler angles or Caley-Klein parameters of the group $SU(2)$. To write down the entries $W_{mn}(q)$, $q\in Sp(1)$, take a formula that calculates $D^{\ell}_{mn}(a,b)$, for example, Equations (13) and (14) from Subsection 4.6.3 of Varshalovich, Moskalev and Khersonskii, Quantum Theory of Angular Momentum, and express $a$ and $b$ in the right hand side in terms of $q$. It is easy to check that
$$
a=\frac{1}{2}(q-\mathrm{i}q\mathrm{i}),\qquad b=-\frac{1}{2}(\mathrm{j}q+\mathrm{k}q\mathrm{i}).
$$ 
