  Formal multiple zeta values allow to study multiple zeta values by algebraic
methods in a way that the open question about their transcendence is
circumvented. In this note we show that Hoffman's basis conjecture for formal
multiple zeta values is implied by the free odd generation conjecture for the
double shuffle Lie algebra. We use the concept of a post-Lie structure for a
convenient approach to the multiplication on the double shuffle group. From
this, we get a coaction on the algebra of formal multiple zeta values. This in
turn allows us to follow the proof of Brown's celebrated and unconditional
theorem for the same result in the context of motivic multiple zeta values. We
need the free odd generation conjecture twice: at first it gives a formula for
the graded dimensions and secondly it is a key to derive a lift of the Zagier
formula to the formal context.
