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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 $\bar k^8$.

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    $\begingroup$ I don't know the exact answer but I can tell you the form it will probably have. Take a cyclic cubic field $L$ over $k$ which is not split (either ramified or inert) at every place $v$ where $i_v \neq 0$. The algebra will be generated by $L$ and $\sigma$ where conjugation by $\sigma$ acts on $L$ as a generator of the Galois group, and $\sigma^3 \in L$ is a particular element. The main game is to choose this element to get the right invariants. $\endgroup$
    – Will Sawin
    Commented Mar 16, 2021 at 15:46
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    $\begingroup$ This will amount to a congruence condition at the finitely many primes with nontrivial invariants and the finitely many ramified primes of $L$, plus the condition that at every non-split unramified prime $v$ with $i_v=0$, the $v$-adic valuation of the element is a multiple of $3$. I don't think one necessarily has a formula for such an element in terms of the invariants so I don't know how explicit one can make it, but one can certainly make it more explicit than this. $\endgroup$
    – Will Sawin
    Commented Mar 16, 2021 at 15:48
  • $\begingroup$ @WillSawin: Thank you, this is very helpful! $\endgroup$
    – Mikhail Borovoi
    Commented Mar 16, 2021 at 17:50
  • $\begingroup$ @MikhailBorovoi: Note that, since all the local invariants are killed by 3, your $D$ has exponent (and then index ) dividing $3$. Hence $D\simeq M_3(D')$, where $D'$ is a central simple algebra of degree $3$. I assume that not all your local invariants are $0$, so $D'$ is actually division. It might simplify your computation... $\endgroup$
    – GreginGre
    Commented Feb 5, 2023 at 22:50

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