A remark on a theorem of Bavula on Lüroth extensions

Using a theorem of Roquette–Ohm [P. Roquette, Isomorphisms of generic splitting fields of simple algebras, J. Reine Angew. Math. 214–215 (1964) 207–226 and J. Ohm, On subfields of rational function fields, Arch. Math. 42 (1984) 136–138], we indicate a short proof that a rational extension of a Lüroth extension of a field k is a Lüroth extension of k. This assertion was proved by Bavula [V. Bavula, Lüroth field extensions, J. Pure Appl. Algebra 199 (2005) 1–10] with the added hypothesis that image.