Pairings and monomorphisms of classifying spaces

We consider the maps between classifying spaces of compact Lie groups of the form BK×BL→BG. If the restriction map BL→BG is a weak epimorphism, then the restriction on BK is known to factor through the classifying spaces of the center of the compact Lie group G. Suppose H is a semi-simple subgroup of a connected compact Lie group G with rank(H)=rank(G). Replacing the weak epimorphism BL→BG by the map BH→BG, analogous results are obtained. We also consider some monomorphisms of classifying spaces of compact Lie groups, such as BSO(n)→BSU(n). Our proof will make use of admissible maps.