A note on the Samelson products in and the group

SO ( 2 n ) has an element α ∈ π 2 n − 1 ( SO ( 2 n ) ) of its homotopy group, which does not pass through SO ( 2 n − 1 ) and vanishes when sent to SO ( 2 n + 1 ) . Thus few results are known about the Samelson product between α and an element in the image from π ∗ ( SO ( 2 n − 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 ( 2 n ) , SO ( 2 n ) ] is not commutative.