Remark

— We observe that formula (6) is valid for every sesquilinear form on $E \times E$ (that is to say, for every function $f$ satisfying (1) and (2), but not necessarily (3)). This remark shows that, when $K = \mathbf{C}$, a sesquilinear form $f$ such that $f(x, x)$ is real for all $x \in E$ is necessarily hermitian: relation (6) then gives $\overline{f(y, x)} = f(x, y)$ since we have $y + \varepsilon x = \varepsilon(x + \overline{\varepsilon} y)$ and $Q(\varepsilon z) = Q(z)$ whenever $\varepsilon^4 = 1$.

From the polarisation formulas, we have in particular,