Remark 4

Since every inverse similitude of $E$ is the product of a direct similitude and the symmetry with respect to $Ae_1$, the matrices No. 1

of inverse similitudes with respect to $(e_1, e_2)$ are the matrices of the form $\begin{pmatrix} a & -\delta b \\ b & -a \end{pmatrix}$.

Henceforth we shall denote by $S$ the group of similitudes of $E$, by $S^+$ that of direct similitudes, by $H$ that of homotheties $\neq 0$, and by $O^+$ that of rotations. Recall that one has $H \subset S^+ (§ 6, n° 5)$.