Let $ G $ denote a classical group.
Question: Is it the case that $$ \langle G_n \otimes G_m,G_m \otimes G_n\rangle=G_{nm} $$ as long as $ n \neq m $?
For example, if $ G $ is the classical group $ GL(\mathbb{C}) $, then the question becomes: is $$ \langle GL_n(\mathbb{C}) \otimes GL_m(\mathbb{C}),GL_m(\mathbb{C}) \otimes GL_n(\mathbb{C})\rangle=GL_{nm}(\mathbb{C}) $$ as long as $ n \neq m $?
If $ n=m $ then we have $ \langle G_n \otimes G_n,G_n \otimes G_n\rangle=G_n \otimes G_n $. And $ G_n \otimes G_n \neq G_{n^2} $ (assuming $ n \neq 1 $).
Interesting to note that, for the connected classical groups, In a compact lie group, can two closed connected subgroups generate a non-closed subgroup? shows that $ \langle G_n \otimes G_m,G_m \otimes G_n\rangle $ must be closed. And I expect it is closed for all classical groups, even the non connected ones.
Cross-posted from MSE: https://math.stackexchange.com/questions/4735161/classical-groups-generated-by-tensor-products-of-subgroups
