Let $k_0$ be a field of characteristic 0, $k/k_0$ be a quadratic extension, and $A/k$ be a central simple algebra over $k$ of dimension $9=3^2$ with an involution of second kind $\sigma$. Then $^\sigma\!(xy)={}^\sigma\!y\cdot{}^\sigma\! x$ and the restriction of $\sigma$ to $k$ is the nontrivial element $\tau\in {\rm Gal}(k/k_0)$.

An element $h\in A$ is called Hermitian if $^\sigma\! h=h$. Let $H\subset A$ denote the vector space (over $k_0$) of all Hermitian elements. Then ${\rm dim}_{k_0}\, H=9$. The group $G=A^\times$ acts on $H$ by $$ g\colon \ h\mapsto g\cdot h\cdot {}^\sigma\!g.$$ I would like to describe all orbits of this action with not too many parameters.

If $A=M_3(k)$ and $^\sigma\! x={}^\tau\! x^T$ (where $\ ^T$ denotes the transpose of a matrix), then any Hermitian matrix is equivalent to a diagonal matrix ${\rm diag}(\lambda_1,\lambda_2,\lambda_3)$ with $\lambda_i\in k_0$. Thus we can describe all orbits (not uniquely) by 3 parameters.

Question. What can be said about a parametrization of the orbits in $H$ in the case when $A$ is a division algebra of dimension $3^2$ over $k$?

Feel free to add relevant tags.