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 $k$?
Motivation: I need a parametrization of $k_0$-forms of the 8-dimensional Lie algebra $\frak{sl}_3$ in order to get a parametrization of $k_0$-forms of the generic trivector on $(k_0)^8$, which can be described in terms of $\frak{sl}_3$. A Hermitian element $h\in A$ defines an algebraic $k_0$-group $G={\rm SU}_1(A,h)$, which is a $k_0$-form of ${\rm SL}_3$, and a Lie algebra ${\rm Lie}\,G$, which is a $k_0$-form of $\frak{sl}_3$.
