Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are defined by the Bourbaki tables.
Denote by $2U_\varphi$ the set of matrices $A^2$ where $A\in U_\varphi$. The symplectic group also has a maximal torus $T$, consisting of the diagonal matrices in $\text{Sp}_{2g}(\mathbb{Z})$.
The kernel $\Gamma(2)$ of the canoncical reduction map $\text{Sp}_{2g}(\mathbb{Z})\to\text{Sp}_{2g}(\mathbb{Z}/2\mathbb{Z})$ is the principal congruence subgroup of level $2$.
The question
I want to show that $\Gamma(2)$ is the group generated by $T$ and the $2U_\varphi$. The inclusion $\langle T,2U_\varphi:\varphi\in C_g\rangle\subset \Gamma(2)$ is trivial, by definition of the root subgroups and because the only nonzero entries of any matrix in $T$ are $1$ and $-1$. How do I prove the other inclusion?