  This article is on the parametrization of the local Langlands correspondence
over local fields for non-quasi-split groups according to the philosophy of
Vogan. We show that a parametrization indexed by the basic part of the Kottwitz
set (which is an extension of the set of pure inner twists) implies a
parametrization indexed by the full Kottwitz set. On the Galois side, we
consider irreducible algebraic representations of the full centralizer group of
the $L$-parameter (i.e not a component group). When $F$ is a $p$-adic field, we
discuss a generalization of the endoscopic character identity.
