I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) Lie group and $R/Z$ is the circle group.) Surely someone has worked this out, but my literature search came up empty.

Alternatively, an explicit representative of the corresponding class in $H^4(BG, Z)$ would be useful, since Cor. 1.14 of Baker 1977 shows how to convert this to an $H^3_{Borel}(G, R/Z)$ representative.

I suspect this is not difficult, but I seem to be stuck, so I'm availing myself to MO.

I'm looking for explicit representatives of $H^3_{Borel}(G, R/Z)$, i.e. a measurable function $G^3\to R/Z$ representing a generator of the cohomology group. (Here $G$ is a compact (perhaps simple) Lie group and $R/Z$ is the circle group.) Surely someone has worked this out, but my literature search came up empty.

Alternatively, an explicit representative of the corresponding class in $H^4(BG, Z)$ would be useful, since Cor. 1.14 of Baker 1977 shows how to convert this to an $H^3_{Borel}(G, R/Z)$ representative.

I suspect this is not difficult, but I seem to be stuck, so I'm availing myself to MO.

[added later] (1) An explicit description of the corresponding Cheeger-Simons differential character would be another welcome alternative answer. (2) In lieu of a general answer, answers for particular groups, e.g. $SU(2)$, would be welcome.