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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?

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    Dieudonné's book on the geometry of classical groups argues that for most fields (and surely for all fields of characteristic zero) the symplectic group is generated by symplectic transvections (and that there is a simple bound on the number of these needed to write an element) It references [Dieudonné, Jean Sur les générateurs des groupes classiques. (French) Summa Brasil. Math. 3 (1955), 149–149.]2011-05-09

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