On Samelson products in p-localized unitary groups

The unitary group U(n) has elements εi∈π2i+1(U(n)) (0⩽i⩽n−1) of its homotopy groups in the stable range. In this paper we show that certain multi Samelson products of type 〈εi,〈εj,εk〉〉 are non-trivial. This leads us to the result that the nilpotency class of the group of the self homotopy set [SU(n),SU(n)] is no less than 3, if 4⩽n. Also by the power of generalized Samelson products, we can see the further result that, for a prime p and an integer n=pk, nil[SU(n),SU(n)](p)⩾3, if (1) p⩾7 or (2) p=5 and n≡0 or 1mod4.