Remark 1

Let $\nu$ be the similarity of matrix $\begin{pmatrix} 0 & \delta \\ 1 & 0 \end{pmatrix}$ with respect to $(e_1, e_2)$; the similarity $\omega$ introduced in the proof of Prop. 1 is equal to $-\alpha_1 \nu$; one has $\nu^2 = \delta$. The multiplier of the direct similarity $u = a + b \nu$ ($a, b$ in $\Lambda$) is equal to the determinant of its matrix $\begin{pmatrix} a & \delta b \\ b & a \end{pmatrix}$, that is to say to $a^2 - \delta b^2 = (a + b \nu)(a - b \nu) = u . \overline{u}$, denoting by $\overline{u}$ the conjugate of $u$ in the algebra $\Lambda(\Phi)$ (Chap. II, § 7, No. 7); in other words the multiplier of $u$ is the norm $N(u)$ of $u$ in the algebra $\Lambda(\Phi)$ (ibid.). In particular, in order that a direct similarity $u$ be a rotation, it is necessary and sufficient that $N(u) = 1$; in order that $u$ be a homothety, it is necessary and sufficient that $u \in \Lambda^*$.