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 $\frak{sl}_3$.