  In this note, we initiate the systematic study of the Lie algebra structure
of the necklace Lie algebra n of a free algebra in 2d variables. We begin by
giving a description of n as an sp(2d)-module. Specializing to d = 1, we
decompose n into a direct sum of highest weight modules for sl_2, the
coefficients of which are given by a closed formula. Next, we observe that n
has a nontrivial center, which we link through the center C of the trace ring
of couples of generic 2x2 matrices to the Poisson center of S(sl_2). The Lie
algebra structure of n induces a Poisson structure on C, the symplectic leaves
of which we are able to describe as coadjoint orbits for the Lie group of the
semidirect product sl_2\rtimes h of sl_2 with the Heisenberg Lie algebra h.
Finally, we provide a link between double Poisson algebras on one hand and
Poisson orders on the other hand, showing that all trace rings of a double
Poisson algebra are Poisson orders over their center.
