  We present results for the QED contributions to the anomalous magnetic moment
of the muon containing closed electron loops. The main focus is on perturbative
corrections at four-loop order where the external photon couples to the
external muon. Furthermore, all four-loop contributions involving
simultaneously a closed electron and tau loop are computed. In combination with
our recent results on the light-by-light-type corrections (see Ref.
\cite{Kurz:2015bia}) the complete four-loop electron-loop contribution to the
anomalous magnetic moment of the muon has been obtained with an independent
calculation. Our calculation is based on an asymptotic expansion in the ratio
of the electron and the muon mass and shows the importance of higher order
terms in this ratio. We perform a detailed comparison with results available in
the literature and find good numerical agreement.
  As a by-product we present analytic results for the on-shell muon mass and
wave function renormalization constants at three-loop order including massive
closed electron and tau loops, which we also calculated using the method of
asymptotic expansion.
