In his book *Introduction to arithmetic groups*, Dave Witte Morris implicitly gives a construction of central division algebras of degree 3 over $\mathbb{Q}$ in Proposition 6.7.4. More precisely, let $L/\mathbb{Q}$ be a cubic Galois extension and $\sigma$ a generator of its Galois group.If $p \in \mathbb{Z}^+$ and $p \neq t\sigma(t)\sigma^2(t)$ for all $t \in L$, then
$$ D=\left\{ \begin{pmatrix}
x & y & z\\
p\sigma(z) & \sigma(x) & \sigma(y)\\
p\sigma^2(y) & p\sigma^2(z) & \sigma^2(x)
\end{pmatrix} :(x,y,z)\in L^3 \right\}
$$
is a division algebra.

On page 145, just before Proposition 6.8.8, Morris claims that it is known that every division algebra of degree 3 arises in this manner. This should follow from the fact that every central division algebra of degree 3 is cyclic. I could not find this explicit construction in my references (e.g. Pierce - *Associative Algebras*, though maybe I missed something) and I would like to know if there is a reference or a quick way to see that this exhausts all central division algebras of degree 3 over $\mathbb{Q}$.