  We investigate the radiative quantum electrodynamic (QED) corrections to the
lepton ($L=e, ~\mu $ and $\tau$) anomalous magnetic moment due to the
contributions of diagrams with insertions of the photon vacuum polarisation
operator consisting solely of four closed lepton ($l=e, ~\mu $ and $\tau$)
loops. Moreover, we focus on specific operators with two loops formed by
leptons $L$ of the same type as the external one, the other two formed by
leptons $\ell$ different from $L$. The approach is essentially based on the
employment of the Mellin-Barnes representation of the $x$-parametrization of
the corresponding Feynman diagrams. This allows one to obtain, for the first
time, exact analytical expressions for the radiative corrections of the tenth
order w.r.t. the electromagnetic coupling constant $e$. Analytically, the
radiative corrections are expressed in terms of the ratio $r=m_\ell/m_L$ of the
internal to external lepton masses. The dependence on $r$ is investigated
numerically in the whole interval of $r$, $0 < r < \infty$. To make comparisons
with earlier published results possible, our exact analytical expressions are
expanded about $r=0$ and $r \to \infty$ and compared with the corresponding
asymptotic expansions known in the literature.
