Ngo famously proved the Langlands-Shelstad fundamental lemma for Lie algebras using the geometry of the Hitchin fibration.

For the purposes of the trace formula, one actually needs the fundamental lemma for the group. In his ICM notes, Ngo states that Waldspurger shows how to reduce the group statement to the Lie algebra statement, but the reference is not clear (to me at least).

My question is

Where is this reduction discussed? A precise reference would be great.

In lieu of the reference, I would also gladly accept a sketch of the reduction:

How does this reduction work?