I am an experienced attorney now engaged exclusively in mediation.  I mediate in my areas of expertise, acquired practicing law for over twenty years in Austin, Texas.  I specialized in civil trial law and was Board Certified in that area by the Texas Board of Legal Specialization.  The emphasis of my practice was construction contracting and litigation.
On Style:  The Objective Inquiry
The Oracle at Delphi was once asked who was the wisest man in Greece, and she replied "Socrates."   When this was conveyed to Socrates he was astounded.  He had little regard for the Greek pantheon of deities, but the priestess he greatly respected, and what she had said was not true.  He knew he was not the wisest man in Greece. But after much questioning of others--as was his habit--and on further reflection, the meaning came into focus, and yes, as a kind of a riddle; that he was the wisest because he knew what it was he did not know.

The mediator does not know anything approaching what the attorneys and parties know collectively, and so must address the situation with humility, and learn.  And the learning is best done by questions that are Socratic, that is, not confrontive--which engenders resistance--but purely on inquiry.  

What can then occur is that in asking the right questions--which eventually both inquire and then also confront--not only does the mediator learn more, but the disputants learn more as well. And, knowing more than they thought they did at the outset--learning what it is they did not​ know--they begin to ask themselves the really challenging questions.

Self-doubt is the beginning of the settlement process.