On the compatibility between cup products, the Alekseev--Torossian connection and the Kashiwara--Vergne conjecture

For a finite-dimensional Lie algebra $\mathfrak{g}$ over a field $\mathbb K\supset \mathbb C$, we deduce from the compatibility between cup products Kontsevich (2003, Section 8) and from the main result of Shoikhet (2001) an alternative way of re-writing Kontsevich product $\star$ on $\mathrm S(\mathfrak g)$ by means of the Alekseev–Torossian flat connection (Alekseev and Torossian, 2010). We deduce a similar formula directly from the Kashiwara–Vergne conjecture (Kashiwara and Vergne, 1978).