A note on the Samelson products in π*(SO(2n)) and the group [SO(2n), SO(2n)]

SO(2n) has an element alpha is an element of pi(2n-1) (SO(2n)) of its homotopy group, which does not pass through SO(2n-1) and vanishes when sent to SO(2n+1). Thus few results are known about the Samelson product between alpha and an element in the image from pi*(SO(2n - 1)). In this paper, we show the non-triviality of certain Samelson products involving alpha and determine for which prime p the p-localization of the group of self homotopy set \SO(2n), SO(2n)\ is not commutative. (C) 2006 Elsevier B.V. All rights reserved.