Given a non-selfconjugate representation $R$ and its conjugate representation $\bar{R}$ of a group $G$, why do we have $$(R \otimes R \otimes \bar{R} \otimes \bar{R})_1= (R \otimes \bar{R} \otimes R\otimes \bar{R})_1 ,$$ where the subscript $1$ means that we only consider the $1$ dimensional part of the tensor product. (In general we have $R \otimes \bar R= 1 \oplus \ldots$ and here we are only interested in the group invariant, i.e. the part of the product that transforms as the one-dimensional representation)
Maybe even the more general result
$$R \otimes R \otimes \bar{R} \otimes \bar{R} = R \otimes \bar{R} \otimes R\otimes \bar{R} ,$$
which I checked explicitly for a few cases.