Corollary 1

The group $S^+$ of direct similitudes is commutative. Whatever the non-isotropic vectors $x, y$ of $E$, there exists one and only one direct similitude $u$ such that $y = u(x)$.

The first assertion follows from the fact that the algebra $A(\Phi)$ is commutative. Since $E$ is a monogenic free $A(\Phi)$-module generated by $x$ (resp. $y$), there exists one and only one element $u$ (resp. $u'$) of $A(\Phi)$ such that $y = u(x)$ (resp. $x = u'(y)$); hence $x = u(u'(x))$, and $uu'$ is the identity; this shows that $u$ is invertible, and is therefore a direct similitude.