Remark

Retain the notation of the preceding proof. Let $c'$ be the $\mathbf{C}$-linear extension of $c$ to $V \otimes_{\mathbf{R}} \mathbf{C}$, and $c''$ the restriction of $c'$ to $P \otimes_{\mathbf{R}} \mathbf{C}$. From our study of $c|P$, $c''$ has an eigenvector $x$ corresponding to the eigenvalue $\exp \frac{2i\pi}{h}$, and this eigenvector does not belong to any of the sets $D \otimes_{\mathbf{R}} \mathbf{C}$, where D denotes a line of P (since D is not stable under c). Now, for any $H \in \mathfrak{H}$, we have seen that $H \cap P$ is a line; thus, $x \notin H \otimes_{\mathbf{R}} \mathbf{C}$.