Fibrewise rational H-spaces

We prove fibrewise versions of classical theorems of Hopf and Leray-Samelson. Our results imply the fibrewise H-triviality after rationalization of a certain class of fibrewise H-spaces. They apply, in particular, to universal adjoint bundles. From this, we may retrieve a result of Crabb and Sutherland [Proc. London Math. Soc. (2000), 747-768], which is used there as a crucial step in establishing their main finiteness result.