If $G_\mathbb{C}$ is a simply-connected simple complex Lie group, theorem 5.4.10 of Brylinski's "Loop Spaces, Characteristic Classes, and Geometric Quantization" claims that there is a natural $\mathcal{O}_{G_\mathbb{C}}^*$-banded gerbe on $G_\mathbb{C}$, essentially corresponding to a generator of $H^3(G_\mathbb{C}, \mathbb{Z})=\mathbb{Z}$; the gerbe comes with some extra "connective" structure. However, he cryptically says on p. 227 of my edition that

an explicit construction of [this gerbe] would involve, in one way or another, algebraic $K$-theory.

He goes on to give an explicit (ad hoc) construction in the case of $SL(2, \mathbb{C})$, but says no more about the algebraic $K$-theory construction. I can't figure out the relationship to algebraic $K$-theory at all, so my question is:

How does one construct this gerbe (with connective structure) via algebraic $K$-theory? I'd also be happy with other constructions, even if they are special to the case of $SL(n, \mathbb{C})$, say.

My interest in this question comes from the fact that it gives a (reasonably) concrete geometric interpretation of the $2$nd Chern class of a principal $G_\mathbb{C}$-bundle (say, $G_\mathbb{C}=SL(n, \mathbb{C})$) as follows: the second Chern class of $P$ may be viewed as the obstruction to building a gerbe with connective structure on the total space of $P$ which is fiberwise equivalent to the canonical gerbe on $G_\mathbb{C}$. Actually, this even gives a geometric interpretation of the "refined" $2$nd Chern class valued in the Deligne cohomology of the base.