We discuss standard pairs of generators of cyclic division p-algebras of degree p, and prove for p = 3 that any two Artin-Schreier elements are connected by a chain of standard pairs. This result has immediate applications to the presentations of such algebras.
Vishne, U. (2002). Generators of central simple p-algebras of degree 3. Israel Journal of Mathematics, 129, 175-187. https://doi.org/10.1007/BF02773162 (Original work published 2002)