If I have two semisimple Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$, then I have read that the irreducible representations of $\mathfrak{g} \oplus \mathfrak{h}$ are precisely the tensor products of the irreducible representations of $\mathfrak{g}$ and $\mathfrak{h}$. Why is this the case?
My question is two-fold. First, if $V$ and $W$ are irreps of $\mathfrak{g}$ and $\mathfrak{h}$ respectively, then why is $V \otimes W$ an irrep of $\mathfrak{g} \oplus \mathfrak{h}$? And second, if $U$ is an irrep of $\mathfrak{g} \oplus \mathfrak{h}$, then how can I decompose it into a tensor product $V \otimes W$ such that $V$, $W$ are irreps of $\mathfrak{g}$, $\mathfrak{h}$ respectively.
