Consider the restricted Lorentz group $SO^+(3,1;\mathbb{R})$ and its complexification $$SO(3,1;\mathbb{C}).\tag{1}$$ Picking the COM frame the massive little group becomes the 3D rotation group $SO(3,\mathbb{R})$.

The double covers are isomorphic to $SL(2,\mathbb{C})$, $$G~:=~SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R\tag{2}$$ and $H:=SU(2)$, respectively, cf. e.g. this Phys.SE post.

The (double cover of the) little group $H$ is a subgroup of $G$ via the diagonal imbedding $$H~\ni~h\quad\mapsto\quad(h,h)~\in~G.\tag{3}$$

If $\rho: G\to GL(V)$ is a group representation for $G$, it induces a restricted representation $\rho|_H:H\to GL(V)$ on the subgroup $H$.

For an irreducible $G$-representation $$(j_L,j_R)~=~j_L\otimes j_R, \qquad j_L, j_R~\in~ \frac{1}{2}\mathbb{N}_0,\tag{4}$$ the restricted $H$-representation is $$(j_L,j_R)~=~j_L\otimes j_R~\cong~\oplus_{j=|j_L-j_r|}^{j_L+j_R} j,\tag{5}$$ as mentioned in OP's title. Note that the notation in expression (4) and (5) are deceptively similar. However we stress that it is implicitly implied that expression (4) is a $G$-representation, while expression (5) is an $H$-representation.

Consider the restricted Lorentz group $SO^+(3,1;\mathbb{R})$ and its complexification $$SO(3,1;\mathbb{C}).\tag{1}$$ Picking the COM frame the massive little group becomes the 3D rotation group $SO(3,\mathbb{R})$.

The double covers are isomorphic to $SL(2,\mathbb{C})$, $$G~:=~SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R\tag{2}$$ and $H:=SU(2)$, respectively, cf. e.g. this Phys.SE post.

The (double cover of the) little group $H$ is a subgroup of $G$ via the diagonal imbedding $$H~\ni~h\quad\mapsto\quad(h,h)~\in~G.\tag{3}$$

If $\rho: G\to GL(V)$ is a group representation for $G$, it induces a restricted representation $\rho|_H:H\to GL(V)$ on the subgroup $H$.

For an irreducible $G$-representation $$(j_L,j_R)~=~j_L\otimes j_R, \qquad j_L, j_R~\in~ \frac{1}{2}\mathbb{N}_0,\tag{4}$$ the restricted $H$-representation is $$j_L\otimes j_R~\cong~\oplus_{j=|j_L-j_r|}^{j_L+j_R} j,\tag{5}$$ as mentioned in OP's title. Note that the notation in expression (4) and (5) are deceptively similar. However we stress that it is implicitly implied that expression (4) is a $G$-representation, while expression (5) is an $H$-representation.

Consider the restricted Lorentz group $SO^+(3,1;\mathbb{R})$ and its complexification $$SO(3,1;\mathbb{C}).$$ Picking the COM frame the massive little group becomes the 3D rotation group $SO(3,\mathbb{R})$.

The double covers are isomorphic to $SL(2,\mathbb{C})$, $$G~:=~SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R$$ and $H:=SU(2)$, respectively, cf. e.g. this Phys.SE post.

The (double cover of the) little group $H$ is a subgroup of $G$ via the diagonal imbedding $$H~\ni~h\quad\mapsto\quad(h,h)~\in~G.$$

If $\rho: G\to V$ is a representation for $G$, it induces a restricted representation $\rho|_H:H\to V$ on the subgroup $H$.

For an irreducible $G$-representation $$(j_L,j_R)~=~j_L\otimes j_R, \qquad j_L, j_R~\in~ \frac{1}{2}\mathbb{N}_0,$$ the restricted $H$-representation is $$(j_L,j_R)~=~j_L\otimes j_R~\cong~\oplus_{j=|j_L-j_r|}^{j_L+j_R} j,$$ as mentioned in OP's title.

Consider the restricted Lorentz group $SO^+(3,1;\mathbb{R})$ and its complexification $$SO(3,1;\mathbb{C}).\tag{1}$$ Picking the COM frame the massive little group becomes the 3D rotation group $SO(3,\mathbb{R})$.

The double covers are isomorphic to $SL(2,\mathbb{C})$, $$G~:=~SL(2,\mathbb{C})_L\times SL(2,\mathbb{C})_R\tag{2}$$ and $H:=SU(2)$, respectively, cf. e.g. this Phys.SE post.

The (double cover of the) little group $H$ is a subgroup of $G$ via the diagonal imbedding $$H~\ni~h\quad\mapsto\quad(h,h)~\in~G.\tag{3}$$

If $\rho: G\to GL(V)$ is a group representation for $G$, it induces a restricted representation $\rho|_H:H\to GL(V)$ on the subgroup $H$.

For an irreducible $G$-representation $$(j_L,j_R)~=~j_L\otimes j_R, \qquad j_L, j_R~\in~ \frac{1}{2}\mathbb{N}_0,\tag{4}$$ the restricted $H$-representation is $$(j_L,j_R)~=~j_L\otimes j_R~\cong~\oplus_{j=|j_L-j_r|}^{j_L+j_R} j,\tag{5}$$ as mentioned in OP's title. Note that the notation in expression (4) and (5) are deceptively similar. However we stress that it is implicitly implied that expression (4) is a $G$-representation, while expression (5) is an $H$-representation.