It depends on how the brackets of the Lie Algebra is being defined.

If it's defined via the adjoint map ad = derivative of Ad, then note that, for all a, Ad_a is the identity for a commutative group.

If it's defined via the exponential map, then use that exp(v1)exp(v2) = exp(v2)exp(v1) for a commutative group, to show that the second order term of f in exp(v1)exp(v2) = exp(f(v1, v2)) that defines the bracket is both symmetric and skew symmetric (and bilinear), hence is 0.