Let $k$ be a global field, for example $k=\Bbb Q$.
Let  $D$ denote the central simple algebra of dimension 9 over $k$ with given local invariants $i_v$. 
Here $v$ runs over the set $\Omega_f(k)$ of finite places of $k$, 
$\, $  $i_v\in \frac13\Bbb Z/\Bbb Z$, $\, $
$i_v=0$ for almost all $v$, $\, $ and $\sum_v i_v=0$. 

>  **Question.** How can one *explicitly* describe the multiplication law in $D$ in terms of the local invariants $i_v$?

**Motivation.** From the multiplication law in $D$, I can obtain the commutation law in the 8-dimensional  Lie algebra ${\frak g}={\frak sl}(1,D)$.
From $\frak g$, I can obtain an explicit trilinear alternating form on the 8-dimensional space $\frak g$:
$$(x,y,z)\mapsto ([x,y],z)\quad\text{for }\,x,y,z\in{\frak g},$$
where $(\,,\,)$ denotes the Killing form. 
This is a $k$-form of a generic alternating trilinear form on $k^8$.