The odd primary order of the commutator on low rank Lie groups

Let G be a simple, simply-connected, compact Lie group of low rank relative to a fixed prime p. After localization at p, there is a space A which “generates” G in a certain sense. Assuming G satisfies a homotopy nilpotency condition relative to p, we show that the Samelson product 〈1G,1G〉 of the identity of G equals the order of the Samelson product 〈ı,ı〉 of the inclusion ı:A→G. Applying this result, we calculate the orders of 〈1G,1G〉 for all p-regular Lie groups and give bounds of the orders of 〈1G,1G〉 for certain quasi-p-regular Lie groups.