Motivated by the following RG question we ask a related question as follows:

We identify $\mathbb{R}^{n} \otimes \mathbb{R}^{m}$ with $\mathbb{R}^{mn}$. We define $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ as a subgroup of $GL(\mathbb{R}^{n} \otimes \mathbb{R}^{m})$ which consist simple tensors $T\otimes S\;\; \text{where}\;\;T\in GL(\mathbb{R}^{n}),\;S\in GL(\mathbb{R}^{m})$.

Is the action of $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ on $\mathbb{R}P^{(mn-1)}$ a transitive action? If not, what is the topological description of the quotion $\mathbb{R}P^{mn-1}/GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$?

The above action is the restriction of the natural action of $GL(\mathbb{R}^{n} \otimes \mathbb{R}^{m})$

Is really $GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$ a proper subgroup of $GL(\mathbb{R}^{n} \otimes \mathbb{R}^{m})$? If not, what is the structure of its Lie algebra?

The later question is somehow similar to the following posts(very indirectly):

positive element in C* tensor product

positive elements in tensor products

Note: "$GL(\mathbb{R}^{n})\otimes GL(\mathbb{R}^{m})$" is just a terminology and notation. We do not confuse it with the tensor product of two groups or some thing else.