Let $V_d$ be a $d+1$-dimensional $k$-vector space with $\text{char} k = 0$. Suppose that $V_d$ is an irreducible representation of the Lie algebra $\mathfrak{sl}_2$ and let $k[V_d]^{\mathfrak{sl}_2}$ be the corresponding algebra of invariants.
Is it true that all irreducible representations of the direct sum $\mathfrak{sl}_2 \oplus \mathfrak{sl}_2$ are isomorphic to the tensor product $V_n \otimes V_m$?
If so, is it true that $k[V_n \otimes V_m]^{\mathfrak{sl}_2 \oplus \mathfrak{sl}_2} \cong k[V_m]^{\mathfrak{sl}_2} \otimes k[V_n]^{\mathfrak{sl}_2}$?
If this is the case, what is the form of this isomorphism?
The first two questions seem quite plausible, but there are some doubts.
Thank you for the answers.