  Motivated by the Langlands' beyond endoscopy proposal for establishing
functoriality, we study the representation $\otimes^3$ in a setting related to
the Langlands $L$-functions $L(s,\pi,\,\otimes^3),$ where $\pi$ is a cuspidal
automorphic representation of $G$ where $G$ is either $\mathrm{SO}(2n+1)$,
$\mathrm{Sp}(2n)$ and $\mathrm{SO}(2n)$. In particular, under what conditions
on partitions $\lambda$, we examine whether or not $\otimes^3$ detects the
subgroups $\mathbb{S}_{[\lambda]}(G)$ for $G$ with type $B_n$ and $D_{2n}$ or
$\mathbb{S}_{\langle\lambda\rangle}(G)$ for $G$ with type $C_n$. Here
$\mathbb{S}_{[\lambda]}$ and $\mathbb{S}_{\langle\lambda\rangle}$ are the usual
Schur functors associated to the partition $\lambda$.
