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. 81 (2000) 747–768], which is used there as a crucial step in establishing their main finiteness result.
Gregory Lupton. Samuel Bruce Smith. "Fibrewise rational H-spaces." Algebr. Geom. Topol. 12 (3) 1667 - 1694, 2012. https://doi.org/10.2140/agt.2012.12.1667