You could check the paper by James York, the one who actually invented it. "Role of conformal three-geometry in the dynamics of gravitation". Physical Review Letters 28 (16): 1082. I haven't looked at it in a long time but I remember York doing a lot of that stuff. You might forward reference it
You may like the derivation in my book: General Relativity (Second Edition), Graduate Texts in Physics, Springer (2013). See especially at Sect. 3.6.2.