The main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(ℤn, G). We show that this stabilised space of homomorphisms decomposes after suspending once with “summands” which can be reassembled, in a sense to be made precise below, into the individual spaces Hom(ℤn, G) after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilised space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilised space also allows the description of the additive reduced homology of the individual spaces Hom(ℤn, G), with the order of the Weyl group inverted.