With the shorthand notation $$H~:=~SU(2)~\subseteq~SL(2,\mathbb{C})~=: ~G,$$

• then $G$ is the double cover of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$,

• and $G_L\times G_R$ is the double cover of the complexified proper Lorentz group $SO(1,3;\mathbb{C})$,

cf. e.g. this related Phys.SE post.

$G$ is isomorphic to the diagonal imbedding $$G~\cong~\{(g,g) \mid g\in G\} ~\subseteq~ G_L\times G_R,$$ and hence a subgroup of $G_L\times G_R$.

OP asks about the $(\frac{1}{2},\frac{1}{2})$ representation $V_L\otimes V_R$, where $V_L$ and $V_R$ denote the left-handed and the right-handed Weyl-spinor representation, respectively, cf. e.g. this related Phys.SE post.

• $V_L$ is the fundamental/defining 2-dimensional representation,

• while $V_R$ is the complex conjugate 2-dimensional representation.

Any representation of a group induces a restricted representation on a subgroup.

$V_L\otimes V_R$ is an irreducible representation wrt. the groups $$ G_L\times G_R, \quad G, \quad H_L\times H_R, $$ but it is an reducible representation $ \frac{1}{2}\otimes\frac{1}{2} \cong 0\oplus 1$ wrt. the subgroup $H$.

This latter fact is related to that the fundamental representation of $H$ is equivalent to the complex conjugate representation, cf. e.g. my Phys.SE answer here. The same does not hold for $G$.

With the shorthand notation $$H~:=~SU(2)~\subseteq~SL(2,\mathbb{C})~=: ~G,$$

• then $G$ is the double cover of the restricted Lorentz group $SO^+(1,3;\mathbb{R})$,

• and $G_L\times G_R$ is the double cover of the complexified proper Lorentz group $SO(1,3;\mathbb{C})$,

cf. e.g. this related Phys.SE post.

$G$ is isomorphic to the diagonal imbedding $$G~\cong~\{(g,g) \mid g\in G\} ~\subseteq~ G_L\times G_R,$$ and hence a subgroup of $G_L\times G_R$.

OP asks about the $(\frac{1}{2},\frac{1}{2})$ representation $V_L\otimes V_R$, where $V_L$ and $V_R$ denote the left-handed and the right-handed Weyl-spinor representation, respectively, cf. e.g. this related Phys.SE post.

• $V_L$ is the fundamental/defining 2-dimensional representation,

• while $V_R$ is the complex conjugate 2-dimensional representation.

Note that $V_L$ and $V_R$ are inequivalent representations of $G$.

Any representation of a group induces a restricted representation on a subgroup.

$V_L\otimes V_R$ is an irreducible representation wrt. the groups $$ G_L\times G_R, \quad G, \quad H_L\times H_R, $$ but it is an reducible representation via Clebsch-Gordan decomposition $$\frac{1}{2}\otimes\frac{1}{2} ~\cong~ 0\oplus 1$$ wrt. the subgroup $H$.

This last fact is related to that the fundamental representation of $H$ is equivalent to the complex conjugate representation, cf. e.g. my Phys.SE answer here. The same does not hold for $G$.