Remark 1

The set L of lines (resp. half-lines) with origin 0 of E is a homogeneous space for the abelian group $S^+/H$ (resp. $O^+$) such that the identity element is the only operator leaving invariant all the elements of L. One may therefore apply to L the formulas of Chapter II, 2nd ed., App. II, No. 1; prop. 8 is thus a particular case of the “parallelogram rule”, and formulas (9) and (10) particular cases of formulas (2) (loc. cit.).