Let $\mathbb{H}$ denote the quaternions. If $(w_1,\ldots,w_n)\in \mathbb{H}^n$ we can write $w_i=z_i+jz_{n+i}$ with complex numbers $z_1,\ldots,z_{2n}$. Now let $M$ be the group of all matrices of the form $$m=\begin{pmatrix} a&0&0\\ 0&A&0\\ 0&0&a \end{pmatrix}$$ in $\operatorname{Mat}(\mathbb{H},n+1)$ so that $A\in\operatorname{Mat}(\mathbb{H},n-1)$ fulfilles $AA^*=A^*A=I_{n-1}$ with the standard involution $^*$ (transpose and conjugate in $\mathbb{H}$) and $|a|^2=1$. Then $M$ acts on $\mathbb{H}^n$ via $m.(w_1,\ldots,w_n)^T=(y_1y_{n+1}^{-1},\ldots,y_ny_{n+1}^{-1})$ where $y=m(w+e_{n+1})$.
I want to determine all homogenous polynomials that are invariant under $M$. More specifically I want to show that the invariant homogenous polynomials $p$ of degree $k$ are contained in $\sum_{p+2q+2t=k}\mathbb{C}x^py^{2q}r^{2t}$ where $x=Re(z_1), y^2=|z_1|^2+|z_{n+1}|^2$ and $r^2=|w_1|^2+\ldots+|w_n|^2=\operatorname{norm}(w)$. I know that since $\begin{pmatrix} 1&0&0\\ 0&A&0\\ 0&0&1 \end{pmatrix}\in M$ for all $A\in\operatorname{Sp}(n-1,\mathbb{H})$ and $\operatorname{Sp}(n-1,\mathbb{H})$ acts transitively on the spheres in $\mathbb{H}^{n-1}$, $p$ is a polynomial in $|w_2|^2+\ldots+|w_n|^2$ and $w_1$ but how to continue?
Thanks in advance for every hint or idea on that :).