jats:titleAbstract</jats:title> jats:pWe prove two theorems about the Malcev Lie algebra associated to the Torelli group of a surface of genus jats:italicg</jats:italic>: Stably, it is Koszul and the kernel of the Johnson homomorphism consists only of trivial jats:inline-formula jats:alternatives <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S2050508623000100_inline1.png" /> jats:tex-math $\mathrm{Sp}2g(Z)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>-representations lying in the centre.</jats:p>