  We prove that Kaletha's local Langlands correspondence for regular
supercuspidal representations gives the classical local Jacquet--Langlands
correspondence due to Deligne--Kazhdan--Vigneras and Badulescu. As in a former
joint paper with Oi, where a similar result is proved for the local Langlands
correspondence for the general linear group, the key ingredients in our proof
are the work of Bushnell--Henniart explicitly describing the local
Jacquet--Langlands correspondence for essentially tame supercuspidal
representations and its reinterpretation due to Tam in terms of
Langlands--Shelstad's \(\zeta\)-data. While, under suitable assumptions, this
result follows from more general theorems either in the recent work of
Fintzen--Kaletha--Spice or that of Chan--Oi, our proof does not require any
assumptions. We also complement a few points on the proof in the former paper
with Oi.
