I am trying to understand a proof in this paper (specifically theorem 5.4). In it, a fact is used that every element of the $W^*$-algebra $A$ is a linear combination of exponential unitaries.

I've tried to reason this fact out, but I don't follow where this fact comes from or why it necessarily holds true. I do know that any element can be written as a sum of four unitaries (as explained here), but I can't seem to find any information about being a sum of specifically exponential unitaries, which is important to the logic of the proof.

Why is this true, and is this known specifically about $W^*$-algebras or is it a more general fact? Any information on this would be appreciated.