generators of the symplectic group In Masoud Kamgarpour's paper "Weil Representations" he uses a set of generators for the symplectic group, referring to a book by R. Steinberg which I do not have access to. If it matters at all, I am working in characteristic zero.
After choosing a symplectic basis, the generators can be written
\begin{equation}
\left(
\begin{array}{cc}
A & 0 \newline
0 & (A^t)^{-1}
\end{array} \right), \
\left(
\begin{array}{cc}
I & B \newline
0 & I
\end{array} \right), \ \text{and} \
\left(
\begin{array}{cc}
0 & I \newline
-I & 0
\end{array} \right),
\end{equation}
where $A$ ranges through invertible matrices and $B$ ranges through symmetric matrices. Does anyone know of a reference or an explanation for this, especially a coordinate-free conceptual and/or geometric one?

This is done in
Symplectic groups By Onorato Timothy O'Meara
and uses the fact that the symplectic group over a field is generated by symplectic transvections.
