Solution
= Solution
Direct matrix multiplication gives
$$
A^2=B^2=C^2=D^2=I,
\qquad
[A,B]=[C,D]=Z,
$$
while every other pair among $A,B,C,D$ commutes. Also $Z=-I$ is central. Consequently every commutator of words in the generators lies in $\langle Z\rangle$, so $G'\subseteq\langle Z\rangle$. Since $Z=[A,B]$, the reverse inclusion holds and
$$
\boxed{G'=\langle Z\rangle.}
$$