Classifying spaces for commutativity of low-dimensional Lie groups

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 $B_{\text{com}}G$.