Let $V$ be a real inner product space of odd dimension and $S∈L(V,V)$ an orthogonal transformation. Prove that there is a vector $v$ such that $S^2(v)=v$.
Hint: Orthogonal transformation in inner product spaces satisfies the following relation
$$ <u,v>=<Su,Sv> = <u,S^TS v> ,$$
and have the property $S=S^T$. Note that, $<u,v>=u^T \, v$.