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 $k^8$.

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 $\bar k^8$.