Let G be a compact connected Lie group and n⩾1 an integer. Consider the space of ordered commuting n-tuples in G⁠, Hom(Zn,G)⁠, and its quotient under the adjoint action, Rep(Zn,G):=Hom(Zn,G)/G⁠. In this article, we study and in many cases compute the homotopy groups π2(Hom(Zn,G))⁠. For G simply connected and simple, we show that π2(Hom(Z2,G))≅Z and π2(Rep(Z2,G))≅Z and that on these groups the quotient map Hom(Z2,G)→Rep(Z2,G) induces multiplication by the Dynkin index of G⁠. More generally, we show that if G is simple and Hom(Z2,G)\mathds1⊆Hom(Z2,G) is the path component of the trivial homomorphism, then H2(Hom(Z2,G)\mathds1;Z) is an extension of the Schur multiplier of π1(G)2 by Z⁠. We apply our computations to prove that if BcomG\mathds1 is the classifying space for commutativity at the identity component, then π4(BcomG\mathds1)≅Z⊕Z⁠, and we construct examples of non-trivial transitionally commutative structures on the trivial principal G-bundle over the sphere S4⁠.