Factorization of Lie group elements

I am reading a paper that does the following:

• Constructs a Lie algebra $$\mathfrak{g}$$ of derivations acting on a certain space (as an infinite direct sum of one-dimensional spaces of derivations)
• Defines $$G$$ to be $$\mathrm{exp}(\mathfrak{g})$$. (In fact, the direct sum defining $$\mathfrak{g}$$ is graded, with each degree finite-dimensional, so $$G$$ is really defined as an inverse limit of exponentiations of finite-dimensional Lie algebras.)
• Points out that $$\mathfrak{g}$$ can be decomposed as a direct sum $$\mathfrak{g}_-\oplus\mathfrak{g}_0\oplus\mathfrak{g}_+$$, with each of the three summands closed under the Lie bracket.
• Asserts that each element $$g\in G$$ can therefore be factored as $$g_-\cdot g_0\cdot g_+$$ with $$g_-\in\mathrm{exp}(\mathfrak{g}_-)$$ and so forth.

So my questions are:

1. Is there an easy way to see why this factorization exists and is unique?
2. If I know a particular group element very explicitly, is there a good formula/procedure for finding the factorization?

(I don't think that the inverse limit in the definition is the hard part. I'm comfortable with how an answer for the finite-dimensional case would get me what I want.)

I guess a more basic question is, to what extent is this actually hard, and to what extent should I just learn more basic Lie theory.

• In case people are curious, the paper is Gross-Hacking-Keel-Kontsevich (arxiv.org/abs/1411.1394) and I'm reading their proof of Theorem 1.17, which is actually a result of Kontsevich-Soibelman. – Nathan Reading Mar 12 at 12:01