Distinct unit-vector stabilizers are not transverse

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.