  Let $F$ be a a non-Archimedean local field of characteristic 0 and $G$ be an
inner form of the general linear group $G^*=\mathrm{GL}_n$ over $F$. We show
that the rectifying character appearing in the essentially tame
Jacquet-Langlands correspondence of Bushnell and Henniart for $G$ and $G^*$ can
be factorized into a product of some special characters, called zeta-data in
this paper, in the theory of endoscopy of Langlands and Shelstad. As a
consequence, the essentially tame local Langlands correspondence for $G$ can be
described using admissible embeddings of L-tori.
