Answer to Question #18660 in Abstract Algebra for john.george.milnor

Question

Let J be a nil ideal in an algebra R over a field k of characteristic 0, and let G be the group 1 + J ⊆ U(R). Show that two elements x, x' ∈ J commute iff exp(x) and exp(x') commute in G, in which case we have exp(x) • exp(x') = exp(x + x') ∈ G.