Orthogonal and special orthogonal Lie groups

Let $F:{\mathrm{GL}}_n(\mathbb{R})\to {\mathrm{Sym}}_n(\mathbb{R})$ be $F(A)=A^{T}A$. Its differential is $d{F}_A(X)=X^{T}A+A^{T}X$. This is surjective: for symmetric $S$, take $X=\frac{1}{2}{A}^{-T}S$. Hence [F3] makes $F^{-1}(I)=O(n)$ an embedded submanifold, and it is a subgroup by [F2].

At $A=I$, the tangent kernel is $X^T+X=0$. It is closed under commutators because $(XY-YX{)}^T=Y^T{X}^T-X^T{Y}^T=-(XY-YX)$ for skew-symmetric $X,Y$.

On $O(n)$, $\det (A{)}^2=\det (A^{T}A)=1$, so determinant takes only the values $1$ and $-1$. Its $1$-fibre is therefore open and closed in $O(n)$ and is an embedded Lie subgroup with the same identity tangent space.

For $n=0$ both groups are the one-point group; for $n=1$, $\mathfrak{s}\mathfrak{o}(1)=0$, $O(1)$ is discrete, and $SO(1)$ is trivial. No interval, endpoint, nondegeneracy beyond invertibility in ${\mathrm{GL}}_n$, metric choice, or biconditional occurs. ${\mathrm{A}\mathrm{C}}_{\omega }$ is propagated only through [F1].