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

   - then $G$ is the double cover of the restricted [Lorentz group](https://en.wikipedia.org/wiki/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](https://physics.stackexchange.com/q/28505/2451) related Phys.SE post.

2. $G$ 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 $G_L\times G_R$.

3. 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](https://physics.stackexchange.com/q/149455/2451) related Phys.SE post. 

   - $V_L$ is the [fundamental](https://en.wikipedia.org/wiki/Fundamental_representation)/defining 2-dimensional representation, 

   - while $V_R$ is the [complex conjugate](https://en.wikipedia.org/wiki/Complex_conjugate_representation) 2-dimensional representation.

4. Any [representation](https://en.wikipedia.org/wiki/Group_representation) of a group induces a [restricted representation](https://en.wikipedia.org/wiki/Restricted_representation) on a subgroup.

5. $V_L\otimes V_R$ 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](https://physics.stackexchange.com/a/781880/2451). The same does not hold for $G$.