Is the congruence 
group $\Gamma(2)$ generated by the upper triangular matrix $(1, 2; 0, 1)$
and the lower triangular matrix $(1, 0; 2, 1)$ or does on need to also
throw in the negation of the identity? To be specific, how do I check that 
the negation of the identity is not a word in the above matrices? 



Generators for congruence group $\Gamma(2)$
Yes, you need to throw in $-I$. Check that the set of all matrices
of the form
$$\left(\begin{matrix}
a&b\\\
c&d
\end{matrix}\right)$$
with $b$ and $c$ even and $a\equiv d\equiv1$ (mod $4$) is a subgroup
of the modular group.


