  In this paper, we show that, under certain assumptions, a parabolic induction
$Ind_B^G\lambda$ from the Borel subgroup $B$ of a (real or $p$-adic) reductive
group $G$ decomposes into a direct sum of the form: \[ Ind_B^G\lambda =
\left(Ind_P^G St_M\otimes \chi_0\right) \oplus \left(Ind_P^G
\mathbf{1}_M\otimes \chi_0\right), \] where $P$ is a parabolic subgroup of $G$
with Levi subgroup $M$ of semi-simple rank $1$, $\mathbf{1}_M$ is the trivial
representation of $M$, $St_M$ is the Steinberg representation of $M$ and
$\chi_0$ is a certain character of $M$. We construct examples of this
phenomenon for all simply-connected simple groups of rank at least $2$.
