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 which yields $F$-algebras of dimension $2^n$, but these algebras are not associative so not central simple in general)
One way to construct Quaternionquaternion 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.
