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

SO(2n) has an element α∈π2n−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 α and an element in the image from π∗(SO(2n−1)). In this paper, we show the non-triviality of certain Samelson products involving α and determine for which prime p the p-localization of the group of self homotopy set [SO(2n),SO(2n)] is not commutative.