Proposition 2

Let s be an endomorphism of V.

(i) If s is a reflection, V is the direct sum of the hyperplane $V_s^+$ and the line $V_s^-$.

(ii) Conversely, assume that V is the direct sum of a hyperplane H and a line D such that $s(x) = x$ and $s(y) = -y$ for $x \in H$ and $y \in D$. Then s is a reflection and $H = V_s^+$, $D = V_s^-$. Finally, D is the image of $1 - s$.

(i) If s is a reflection, $V_s^+$ is a hyperplane. If x belongs to $V_s^+ \cap V_s^-$, then $x = s(x) = -x$, so $x = 0$ since K is of characteristic $\neq 2$. Finally, for $x$ in V, the vector $x' = s(x) + x$ (resp. $x'' = s(x) - x$) belongs to $V_s^+$ (resp. $V_s^-$) since $s^2 = 1$, and $2x = x' - x''$. Thus V is the direct sum of $V_s^+$ and $V_s^-$, and $V_s^-$ is necessarily of dimension 1 since $V_s^+$ is a hyperplane.

(ii) Under the stated hypotheses, every element of V can be written uniquely in the form $v = x + y$ with $x \in H$ and $y \in D$ and we have $s(v) = x - y$; assertion (ii) follows immediately from this.