Why is the Haar measure of a Lie group with finite abelianization both left and right translation invariant? (Moved from math.SE)

Question

I'm reading Foundations of Hyperbolic Manifolds by Ratcliffe. On the way to proving Gromov's theorem on the proportionality of hyperbolic volume and simplicial volume, he states that "it is a basic fact of the theory of Haar measure that the Haar measure on a group is both left- and right-invariant if the abelianization of the group is finite." Why should this be?

Answer 1

The modular quasicharacter (quotient of left and right haar measures) is a group homomorphism from G to the positive reals (under multiplication). This obviously factors through the abelianisation since the target is abelian, and there are no homomorphisms from a finite abelian group to the positive reals.