The tensor product of a representation $R$ of a Lie group $G$ with itself is in general a reducible representation of $G$, i.e. a sum of irreducible representations
$$ R \otimes R = R^1 \oplus R^2 \oplus \ldots $$
Under a subgroup $H \subset G$, we have $R = r_1 \oplus r_2 \oplus \ldots$, $R^1= r_1^1 \oplus r_2^1 \oplus \ldots$ and $R^2= r_1^2 \oplus r_2^2 \oplus \ldots$, where $r_i$ denote irreducible representations of $H \subset G$.
If we consider the subspace $R_1$, i.e. the projection of $ R \otimes R$ onto $R_1$, denoted by $(R \otimes R )_{R_1}$, how can I compute the corresponding decomposition under the subgroup $H$? This means I'm looking for a method to compute
$$ R_1 = (R \otimes R)_{R_1} \rightarrow r_1^1 \oplus r_2^1 \oplus \ldots = ( (r_1 \oplus r_2 \oplus \ldots) \otimes (r_1 \oplus r_2 \oplus \ldots))_{r_1^1 \oplus r_2^1 \oplus \ldots} $$
In other words: How can I compute which tensor products of irreducible representations of $H$ appear on the right-hand side, if we restrict to a given subspace $r_1^1 \oplus r_2^1 \oplus \ldots $ of $H$.