I'm trying to prove that, for a standard unimodular even lattice $\Lambda$ (by standard I mean that it is direct sum of copies of the hyperbolic plane $U$ and $E_8$) every element of $O^+(\Lambda)$, i.e. the elements of the orthogonal group of $\Lambda$ with spinor norm $1$ can be written as product of reflections $r_w$ ($r_w(\epsilon)=\epsilon-\frac{2(\epsilon,w)}{(w,w)}w$) with $(w,w)=-2$.
Obviously to define the spinor norm of $g\in O(\Lambda)$ I use the fact that I can write $g$ as a product of reflection $r_{w_1}\cdots r_{w_k}$ and then I calculate the spinor norm $sp(g)$ of $g$ as the product of the spinor norms of the $r_{w_i}$ ($sp(r_{w_i})=1$ if $(w_i,w_i)<0$ and $sp(r_{w_i})=-1$ if $(w_i,w_i)>0$).
In this article of C.T.C. Wall http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002180294&IDDOC=253592 it is proved ( theorem 4.8) that every element of $O(\Lambda)$ can be written as a product of reflections $r_w$ with $(w,w)=\pm 2$. It is close to the statement I'm trying to prove, but it is not sufficient: given Wall's result, for $g\in O^+(\Lambda)$ it may happen $g=r_{w_1}\cdots r_{w_i}r_{w_{i+1}}\cdots r_{w_k}$ with $(w_i,w_i)=(w_{i+1},w_{i+1})=2$ and $(w_j,w_j)=-2$ for $j\neq i,i+1$. So somehow I have to write the product $r_{w_i}r_{w_{i+1}}$ as a product of reflections $r_{u_k}$ with $(u_k,u_k)=-2$.
Does someone have some ideas?
