Basis of the free one-generated transposed Poisson algebra

Determine a linear basis for the free transposed Poisson algebra generated by a single element over a field of characteristic zero, giving an explicit description of a spanning set and proving linear independence.

Background

Transposed Poisson algebras combine an associative-commutative product with a Lie bracket subject to a compatibility identity. While various structural results are known, even the simplest free case—one generator—lacks an explicit basis.

An explicit basis for the one-generated case would be a stepping stone toward understanding free transposed Poisson algebras of higher rank.