Are there any central simple algebras admitting a standard basis?


By a standard basis I mean a normal basis that has a cyclic property generalizing that of the familiar basis $1, i, j, k$ for Quaternion algebras $(a, b \mid F)$, satisfying relations $i^2=a, j^2=b, k=ij=-ji$ for $a, b \in F^\times$.


I saw [Cayley–Dickson construction](https://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction) which yields $F$-algebras of dimension $2^n$, but these algebras are not associative so not central simple in general)


One way to construct quaternion algebras is to use abelian varieties:
Like in the case of elliptic curves, by using $\ell$-adic Tate modules we have the following:



> 
> the endomorphism ring $\text{End}(A)$ is finitely generated $\mathbb{Z}$-module of rank at most $4g^2$.
> 
> 
> 


In general, the endomorphism ring $\text{End(A)}$ of an abelian vairety $A$ is an order in a semi-simple algebra over $\mathbb{Q}$. When $B$ is a semi-simple algebra over $\mathbb{Q}$ admitting a positive definite anti-involution, such algebras have been classified by A. Albert and G. Scorza.


Type I: Totally real number field   

Type II/III: Definite or indefinite quaternion algebra over totally real field   

Type IV: Central simple algebra over a CM-field.


Central simple algebras of dimension $n^2$ are all quaternion algebras when $n=2$. I was wondering if other form of basis with cyclic property appears when $n$ is larger, but at least in the construction above only quaterinon algebras appear.



Are there any central simple algebras admitting a standard basis?
As suggested by [@Kimball](https://mathoverflow.net/questions/414268/are-there-any-central-simple-algebras-admitting-a-standard-basis/414750#comment1063039_414268) I develop my [comment](https://mathoverflow.net/questions/414268/are-there-any-central-simple-algebras-admitting-a-standard-basis/414750#comment1062278_414268). An important class of central simple algebras consists of the *cyclic algebras*: assume that the field $k$ contains a primitive $n$-th root of unity $\zeta $, with $n$ prime to $\operatorname{char}(k) $. Let $a,b\in k^\*$; the cyclic algebra $R\_{a,b}$ is the quotient of $k\langle x,y \rangle$ by the relations $x^n=a, y^n=b, yx=\zeta xy$. The monomials $x^{i}y^{j}$ for $0\leq i,j<n$ form a natural basis of $R\_{a,b}$ over $k$, generalizing the standard basis of the quaternions. Over a number field, every central simple algebra is
cyclic — this is a deep result of Albert–Hasse–Brauer–Noether. It is a classical conjecture that for $n$ prime this should hold over any field.


