Remark 2

In the definition of the group of angles of lines, one may, instead of the group $S^+/H$, use the group $O^+/ \{ -1, 1 \}$ which is canonically isomorphic to it (Prop. 7). The canonical homomorphism of $O^+$ onto $O^+/ \{ -1, 1 \}$ thus corresponds to a homomorphism of $\mathfrak{A}$ onto $\mathfrak{A}_0$, namely the one which associates with the angle of the two half-lines $\Delta, \Delta'$ the angle of the lines $D, D'$ containing respectively $\Delta, \Delta'$. In the case where the field A is the field of real numbers, the group $\mathfrak{A}$ is thus a covering of order 2 of the group $\mathfrak{A}_0$

According to Prop. 8, all angles of lines (resp. half-lines) of the form $(\widehat{D', D''})$ where $D'$ and $D''$ are orthogonal (resp.

of the form ($\widehat{D, - D}$) are equal: this follows in fact from Remark 2 of No. 1 (resp. is obvious). This angle of lines (resp. of half-lines) is called the right angle (resp. the flat angle); it is an element of order 2 of $\mathfrak{A}_0$ (resp. of $\mathfrak{A}$).