  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.
