  Let X be an ordered alphabet. Lie_2(n) (and P_2(n) respectively) are the
multilinear parts of the free Lie algebra (and the free Poisson algebra
respectively) on X with a pair of compatible Lie brackets. In this paper, we
prove the dimension formulas for these two algebras conjectured by B. Feigin by
constructing bases for Lie_2(n) (and P_2(n)) from combinatorial objects. We
also define a complementary space Eil_2(n) to Lie_2(n), give a pairing between
Lie_2(n) and Eil_2(n), and show that the pairing is perfect.
