I am getting slightly confused about the nature of $\times$ in Lie algebras and representations in general. From what I can tell $\times$ is used by some authors to represent both the Cartesian product or the tensor product (in the context of Matrix Lie groups). For example in the formula:
$$\mathcal{L}(G\times G)= \mathcal{L}(G)\oplus\mathcal{L}(G)$$ Is the $\times $ a Cartesian product or a tensor product? Is there even some sort of isomorphism between them so that this ambiguity doesn't actually matter?
I have literally never seen the symbol $\times$ used to denote a tensor product. I would assume that in the equation you wrote, $G$ is a Lie group and $\mathcal{L}(G)$ denotes the Lie algebra of $G$, in which case $G\times G$ definitely refers to the Cartesian product of $G$ with itself, which is again a Lie group (there is no such thing as a tensor product of Lie groups!).