  In this note we show that the second homotopy group of $B(2,G)$, the
classifying space for commutativity for a compact Lie group $G$, contains a
direct summand isomorphic to $\pi_1(G)\oplus\pi_1([G,G])$, where $[G,G]$ is the
commutator subgroup of $G$. It follows from a similar statement for $E(2,G)$,
the homotopy fiber of the canonical inclusion $B(2,G)\hookrightarrow BG$. As a
consequence of our main result we obtain that if $E(2,G)$ is 2-connected, then
$[G,G]$ is simply-connected. This last result completes how the higher
connectivity of $E(2,G)$ resembles the higher connectivity of $[G,G]$ for a
compact Lie group $G$.
