Let D be a division algebra of degree 3 over a field containing a primitive cube root of unity. We give two proofs of a theorem of Rost asserting that any two Kummer elements in D can be connected by a chain of length 4.
Haile, D., Kuo, J.-M., & Tignol, J.-P. (2009). On chains in division algebras of degree 3. Comptes rendus - Mathématique, 347(15-16), 849-852. https://doi.org/10.1016/j.crma.2009.06.007 (Original work published 2009)