The homotopy groups of any group-like space are equipped with a Samelson product satisfying, up to sign, the identities of a graded Lie bracket. We shall compute the Samelson product in two kinds of spaces of selfhomotopy equivalences arising when adding a homotopy or a homology group to a space.
First, let A→ X be a cofibration with a Moore space M(G,n) as cofibre. For the monoid autA (X) of maps under A homotopic (rel. A) to the identity, the Samelson product is a pairing
πn+i(G;X)⨂πn+j(G;X) → π
of homotopy groups with coefficients  in G. Theorem 2.1 computes this pairing in terms of a homomorphism associated to a α ∈ πi(autAX)).