Adjoint action of a finite loop space. II

Adjoint actions of compact simply connected Lie groups are studied by Kozim a and the second author based on the series of studies on the classificatio n of simple Lie groups and their cohomologies. At odd primes, the first aut hor showed that there is a homotopy theoretic approach that will prove the results of Kozima and the second author for any 1-connected finite loop spa ces. In this paper, we use the rationalization of the classifying space to compute the adjoint actions and the cohomology of classifying spaces assumi ng torsion free hypothesis, at the prime 2. And, by using Browder's work on the Kudo-Araki operations Q(1) for homotopy commutative Hopf spaces, we sh ow the converse for general 1-connected finite loop spaces, at the prime 2. This can be done because the inclusion j : G --> B Lambda G satisfies the homotopy commutativity for any non-homotopy commutative loop space G.