Remark 2

The direct similitudes $u = a + b \nu$ ($a, b$ in $\Lambda$, $b \neq 0$) whose square is a homothety are the homotheties and the scalar multiples $b \nu$ of $\nu$, since $(a + b \nu)^2 = (a^2 + \delta b^2) + 2ab \nu$. The latter are none other than the automorphisms of the vector space $E$ which transform every vector into a vector orthogonal to it; in fact, the matrix of such an automorphism is necessarily of the form $\begin{pmatrix} 0 & c \\ d & 0 \end{pmatrix}$ ($c, d$ in $\Lambda$), and the condition that the vector $\lambda e_1 + \mu e_2$ be transformed into an orthogonal vector is then written $\lambda \mu (d \alpha_2 + c \alpha_1) = 0$.