# Basic question regarding symplectic reflections

I am learning about symplectic reflection algebras, following the paper by Etingof and Ginzburg. Given a symplectic $$\mathbb{C}$$-vector space $$V$$ with symplectic (i.e. non-degenerate alternating bilinear) form $$\omega:V\times V\rightarrow \mathbb{C}$$, then $$s$$ is a symplectic reflection if firstly $$s\in Sp(V)$$ (i.e. $$\omega(s(x),s(y))=\omega(x,y)\ \forall x,y\in V$$), and secondly $$rank(1-s)=2$$.

This is likely very obvious, but in the paper it is stated that the direct sum decomposition $$V=im(1-s)\oplus ker(1-s)$$ is $$\omega$$-orthogonal. Which I take to mean for each $$x\in im(1-s)$$ $$\omega(x,y)=0\ \forall y\in ker(1-s)$$. How would one go about proving this?

$$x$$ can be written as $$u-s(u)$$ and $$s(y)=y$$ then $$\omega(x,y) = \omega(u-s(u),y) = \omega(u,y)-\omega(s(u),y) = \omega(u,y)-\omega(s(u),s(y)) = \omega(u,y)-\omega(u,y) =0$$