  Let $F$ be a non-archimedean locally compact field. We study a class of
Langlands-Shahidi pairs $({\bf H},{\bf L})$, consisting of a quasi-split
connected reductive group $\bf H$ over $F$ and a Levi subgroup $\bf L$ which is
closely related to a product of restriction of scalars of ${\rm GL}_1$'s or
${\rm GL}_2$'s. We prove the compatibility of the resulting local factors with
the Langlands correspondence. In particular, let $E$ be a cubic separable
extension of $F$. We consider a simply connected quasi-split semisimple group
$\bf H$ over $F$ of type $D_4$, with triality corresponding to $E$, and let
$\bf L$ be its Levi subgroup with derived group ${\rm Res}_{E/F} {\rm SL}_2$.
In this way we obtain Asai cube local factors attached to irreducible smooth
representations of ${\rm GL}_2(E)$; we prove that they are Weil-Deligne factors
obtained via the local Langlands correspondence for ${\rm GL}_2(E)$ and tensor
induction from $E$ to $F$. A consequence is that Asai cube $\gamma$- and
$\varepsilon$-factors become stable under twists by highly ramified characters.
