Let $\mathbb{A}^{m\times n}$ denote the set of all $m \times n$ matrices with entries in the set $\mathbb{A}$. For a matrix $M$ we let ${^tM}$ denote its transpose, and $M^{-1}$ its inverse, if it is invertible. We denote the $r \times r$ identity matrix by $I_r$.
We define the symplectic modular group of degree $g$ by $$\Gamma^g:=\text{Sp}_{2g}(\mathbb{Z})= \left\{ M \in \mathbb{Z}^{2g \times 2g} \mid {^tM}JM = J \right\} , J = \left(\begin{smallmatrix} {0} & {I_g}\\ {-I_g} & {0} \end{smallmatrix} \right). $$
We define the following subgroup $$\Gamma^g(1,2):= \left\{ M = \left(\begin{smallmatrix} {A} & {B}\\ {C} & {D} \end{smallmatrix} \right) \in \Gamma^g \mid {^tA}C, {^tB}D \text{ have even diagonals} \right\}.$$
Mumford (see Appendix to Chapter 2 Section 5 of Tata Lectures on Theta I) gives the following generators for $\Gamma^g(1,2)$: $$J, \quad \left\{ \left(\begin{smallmatrix} {A} & {0}\\ {0} & {^tA^{-1}} \end{smallmatrix} \right) \mid A \in \text{GL}(g, \mathbb{Z}) \right\}, \quad \left\{ \left(\begin{smallmatrix} {I_g} & {B}\\ {0} & {I_g} \end{smallmatrix} \right) \mid B \text{ is symmetric, integral and has even diagonal} \right\}.$$
Consider the matrix $$E= %\left( \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0 \end{pmatrix} %\right) \in \Gamma^2(1,2).$$ Can anyone tell me how $E$ can be expressed in terms of Mumford's generators?
