Lemma 1

Suppose that $A$ is a field and $\Phi$ a Hermitian form $\neq 0$ on $E$. If all the vectors of $E$ are isotropic, $A$ is a commutative field of characteristic 2, the antiautomorphism $J$ is the identity and $\Phi$ is alternating.

Indeed, on expanding $\Phi(x + y, x + y) = 0$, taking into account the hypotheses $\Phi(x, x) = \Phi(y, y) = 0$, we obtain the relation $\Phi(x, y) = -\overline{\Phi(x, y)}$ for arbitrary $x, y$ in $E$. Since $\Phi$ is $\neq 0$, there exist $x, y$ in $E$ such that $\Phi(x, y) = 1$. Writing that $\Phi(\lambda x, y) = -\Phi(\lambda x, y)$, we obtain $\overline{\lambda} = -\lambda$ for every $\lambda \in A$. Taking first $\lambda = 1$, we see that $A$ is of characteristic 2; the relation $\overline{\lambda} = -\lambda$ then shows that $J$ is the identity, hence $A$ is commutative and $\Phi$ is alternating.