Lemma 1

Let E be a complex vector space and $u$ a semilinear involution on E. Let $x\in E$; put

The pair $(x_1, x_2)$ is the unique element of $E^u\times E^u$ such that $x=x_1+ix_2$.

The elements $x_1$ and $x_2$ satisfy $x_1+ix_2=x$ and belong to $E^u$ since $u(u(x)) =x$. Conversely, if $y_1$ and $y_2$ in $E^u$ satisfy $x=y_1+iy_2$, it follows that $u(x) =u(y_1) +u(iy_2) =y_1-iy_2$, hence