$\mathrm{GL}(n,F)$ as a subgroup of $\mathrm{O}(n,n,F)$ I have stumbled on what to me is an interesting observation: over any commutative ring $F$, the general linear group is a subgroup of the split orthogonal group of twice the dimension:
$$\mathrm{GL}(n,F) \lt \mathrm{O}(n,n,F)$$
This is in a sense the reverse of the more obvious subgroup relation
$$\mathrm{O}(n,n,F) \lt \mathrm{GL}(2n,F)$$
in this case the dimension being the same.
My proof is simple enough – request this if needed.
My questions:

*

*Is my observation true?

*Is this well-known, and in what form would it normally be stated in texts (e.g. as a more general statement)?

EDIT: This is shown by considering the matrix embedding $$\mathrm{M}_n(F)\to\mathrm{M}_{2n}(F):T\mapsto\left[\begin{matrix}T & 0 \\ 0 & (T^{-1})^\mathrm{t}\end{matrix}\right]$$ restricted obviously to invertible $T$, with ${}^\mathrm{t}$ denoting the transpose.
 A: Here is some context in which to place these observations. To my mind, the most "conceptual" maps between groups are those which arise from restricting functors to automorphisms. A good example is the pair of embeddings
$$GL_n(\mathbb{C}) \to GL_{2n}(\mathbb{R})$$
and
$$GL_n(\mathbb{R}) \to GL_n(\mathbb{C})$$
which are somewhat analogous to the pair of embeddings you describe. These correspond to the restriction of scalars / underlying real vector space functor from complex to real vector spaces, resp. the complexification functor $V \mapsto V \otimes_{\mathbb{R}} \mathbb{C}$ from real to complex vector spaces. The significance of looking specifically at the induced maps on automorphisms is that they are related to e.g. the corresponding underlying real vs. complexification functors for real and complex vector bundles. 
The analogous statements for the split orthogonal group are the following. Let's define a quadratic space to be a vector space equipped with a quadratic form. There is a natural forgetful functor from quadratic spaces to vector spaces which is responsible for the "obvious" embeddings of various orthogonal groups into various general linear groups, including for the split orthogonal groups, which correspond to split quadratic spaces.
On the other hand, if $V$ is a vector space, then from it we can construct a quadratic space with underlying vector space $V \oplus V^{\ast}$ and quadratic form
$$q(v \oplus f) = f(v).$$
This quadratic space is split, and this construction is functorial with respect to automorphisms (but, unlike our previous example, not with respect to arbitrary maps). This functor corresponds to your embedding of the general linear group into a split orthogonal group. 
