  We study electronic, charge, and magnetic properties of twisted bilayer
graphene with fillings $2\leq n\leq 6$ per moire unit cell within the recently
introduced formulation of extended dynamical mean-field theory (E-DMFT) for
two-sublattice systems. We use previously obtained hopping parameters between
the states, described by Wannier functions, centered at the lattice spots of AB
and BA stacking, and the long-range Coulomb interaction, obtained within cRPA
analysis. We show that account of spin exchange between AB and BA nearest
neighbour spots is crucial to introduce charge and spin correlations between
these spots. Account of this exchange yields preferable concentration of
electrons in the same valley, with the tendency of parallel spin alignment of
electrons in AB and BA spots, in agreement with earlier results of the strong
coupling analysis, suggested SU(2)$\times$SU(2) emergent spin-valley symmetry.
The local spectral functions show almost gapped state at the fillings
$n=2,4,6$, and incoherent metal state for the other fillings. We find that in
both cases the local states of electrons have rather long lifetime. At the same
time, the non-local charge- and spin susceptibilities, obtained within the
ladder approximation, are peaked at incommensurate wave vectors, which implies
that the above discussed ordering tendencies are characterised by an
incommensurate pattern.
