A free associative algebra as a free module over a Specht subalgebra

Let $k$ be a field of characteristic 0 and let $k\langle X\rangle$ be a free associative algebra with finite basis $X$. Let $R=R(k,X)$ be the universal enveloping algebra of the square of $\operatorname{Lie}(X)$, regarded as a subalgebra of $k\langle X\rangle$ and called the Specht subalgebra of the free algebra. We prove that $k\langle X\rangle$ is a free (left) $R$-module, find sufficient conditions for some system of elements in $k\langle X\rangle$ to be a basis for this module, and obtain an explicit formula that allows us to calculate the $R$-coefficients of the elements of the free algebra over a special basis of “symmetric monomials”.