Extending transvections/generating the symplectic group The context is showing that the symplectic group is generated by symplectic transvections.
At the very bottom of http://www-math.mit.edu/~dav/sympgen.pdf it is stated that any transvection on the orthogonal space to a hyperbolic plane (a plane generated by $u,v$ such that $(u,v)=1$ with respect to the bilinear form) can be extended to a transvection on the whole space with the plane contained in its fixed set.
Is there an easy way to see why this is true? If not, does anyone have a reference/solution?
Thanks.

Assume that $V$ is whole space with $2n$ dimensional. Then the orthogonal space you mentioned (let it be $W$) is $2n-2$ dimensional symplectic vector space (It is very easy to check the conditions). Then take another space spanned by $v$ and $w$ such that $v$, $w$ $\in W$ and $\omega(v,w)=1$. The new space orthogonal to this space is symplectic vector space with dimensional $2n-4$. You know the rest.
