Corollary 2

The group $O^+$ of rotations is commutative. Whatever the vectors $x, y$ of $E$ such that $\Phi(x, x) = \Phi(y, y) \neq 0$, there exists one and only one rotation $u$ such that $y = u(x)$.

The first assertion follows from Corollary 1. The latter also shows that there exists one and only one direct similarity $u$ such that $y = u(x)$; since $\Phi(u(x), u(x)) = \Phi(x, x)$, the multiplier of $u$ is equal to 1, and $u$ is therefore a rotation.