  For each of the groups $G = O(2), SU(2), U(2)$, we compute the integral and
$\mathbb{F}_2$-cohomology rings of $B_\text{com} G$ (the classifying space for
commutativity of $G$), the action of the Steenrod algebra on the mod 2
cohomology, the homotopy type of $E_\text{com} G$ (the homotopy fiber of the
inclusion $B_\text{com} G \to BG$), and some low-dimensional homotopy groups of
$B_\text{com} G$.
