= Projective image of the square symmetry group
{title2=$D_8/\{\pm I\}\cong\mathbb Z_2\times\mathbb Z_2$}
Let $A=\operatorname{diag}(1,-1)$ and $B=\begin{pmatrix}0&1\\1&0\end{pmatrix}$. They generate the <dihedral group> of a square, realized by $\{\pm I,\pm A,\pm B,\pm AB\}$. Its image in the <projective linear group> has four elements, since opposite <matrices> become equal. The projective classes of $A,B$ commute because $AB=-BA$, and each squares to the identity. Sending $[A^aB^b]$ to $(a,b)$ gives the <Klein four-group>. This is an example in which a nontrivial scalar commutator disappears after passing to a <projective representation>.
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