A note on Samelson products and mod p cohomology of classifying spaces of the exceptional Lie groups

Let G be an exceptional Lie group G2, F4, E6, E7 or E8, and also set p is the corresponding prime 7, 13, 13, 19 or 31 respectively. If we localize spaces at p, G can be decomposed into a product of spheres. Using this decomposition, we take some elements in the homotopy groups of p-localized G, and we offer some non-zero 3-fold Samelson products of them. This implies that the nilpotency class of the localized self-homotopy group of G is greater than or equal to 3.The key lemma for these results is about a calculation on the cohomology operator P1 in the cohomology of BG, where G and p are as above. During this calculation, we use some original ideas, which are also used in Kishimoto and Kaji (in press) [7] recently.