Proposition 2

Let $\Phi$ be a positive hermitian form on $E$. For $x, y$ in $E$, one has

The inequality is indeed immediate when the vectors $x$ and $y$ are proportional. We may therefore suppose that $x$ and $y$ are linearly independent. Let $A'$ be the (commutative) subfield $K(\Phi(x, y))$ of $A$, $F$ the vector plane over $A'$ generated by $x$ and $y$, and $\Phi_F$ the restriction of $\Phi$ to $F$; the latter takes its values in $A'$. By Proposition 1, the discriminant of $\Phi_F$ with respect to the basis $(x, y)$ is $\geqslant 0$. Now this discriminant is $\Phi(x, x)\Phi(y, y) - \Phi(x, y)\overline{\Phi(x, y)}$. Q.E.D.