Unit-vector stabilizer in a special orthogonal group ($S_v$)

For $v\in S^n$, the stabilizer $S_v=\{R\in SO(n+1):Rv=v\}$ is conjugate to the block subgroup $SO(n)$ acting on $v^{\perp }$. Hence it is a submanifold of dimension $n(n-1)/2$, with

For distinct $v,w\in S^n$, the submanifolds $S_v,S_w\subseteq SO(n+1)$ are not transverse. They both contain the identity. If $v,w$ are independent, the bivector $v\wedge w$ is a nonzero normal direction common to both tangent spaces there; if $w=-v$, the two stabilizers coincide.