On the characters and commutators of finite algebra groups

Let A be a finite-dimensional algebra over a finite field and let J(A) be the Jacobson radical of A. We prove a new commutator identity in the group G=1+J(A) and using this identity we show that every irreducible character of G is induced from a linear character of some subgroup of the form H=1+U, where U is a subalgebra of J(A). This answers a question of I.M. Isaacs.