$\underline{\text{Question, version 2.0}}$
Sorry again for the confusion, my notation appears to have been sloppy. And sorry that I haven't posted the entire thing on this page, but I really think it's more efficient (for everyone) that I post it here:
http://homepages.utoledo.edu/mcrumle/Reference%20Needed%20-%20Direct%20Products.pdf
This theorem I believe is much more basic than the gracious people who bothered to try to answer it (in my garbled terminology) have made it out to be. I am talking about arbitrary representations (not irreducible, etc.) over arbitrary fields (not algebraically closed, etc.) for arbitrary algebraic groups (not reductive, etc.). It is simply the statement that any $G \times H$-module structure on a $k$-vector space $V$ can be factored (as a product of matrices, or linear maps if you like) uniquely into a product of commuting representations for $G$ and $H$ on that same vector space. In section 2 there are no tensor products or direct sums of vector spaces taking place; the vector space $V$ is fixed throughout, and all three of the groups $G$, $H$, and $G \times H$ are acting on it.
The corollaries that follow in section 3 merely re-affirm the suspicion that, for fixed $G \times H$-representations on the fixed vector spaces $V$ and $W$, things like "tensor product", "direct sum", and "morphism" of $G \times H$-modules should behave as expected with respect to their respective $G$ and $H$ modules.
Thank you once again for any help.
