  In this article we consider a space B_{com}G assembled from commuting
elements in a Lie group G first defined in [Adem, Cohen, Torres-Giese 2012]. We
describe homotopy-theoretic properties of these spaces using homotopy colimits,
and their role as a classifying space for transitionally commutative bundles.
We prove that ZxB_{com}U is a loop space and define a notion of commutative
K-theory for bundles over a finite complex X which is isomorphic to
[X,ZxB_{com}U]. We compute the rational cohomology of B_{com}G for G equal to
any of the classical groups U(n), SU(n) and Sp(n), and exhibit the rational
cohomologies of B_{com}U, B_{com}SU and B_{com}Sp as explicit polynomial rings.
