The Galois coaction on the electron anomalous magnetic moment

Recently S. Laporta published a partial result on the fourth order QED contribution to the electron anomalous magnetic moment $g-2$. This result contains explicit polylogarithmic parts with fourth and sixth roots of unity. In this note we convert Laporta’s result into the motivic `$f$ alphabet’. This provides a much shorter expression which makes the Galois structure visible. We conjecture the $Q$ vector spaces of Galois conjugates of the QED $g-2$ up to weight four. The conversion into the $f$ alphabet relies on a conjecture by D. Broadhurst that iterated integrals in certain Lyndon words provide an algebra basis for the extension of multiple zeta values by sixth roots of unity. We prove this conjecture in the motivic setup.