On free generators in the Grothendieck-Teichmüller Lie Algebra

We prove that any homogeneous family ${\sigma }_3,{\sigma }_5,\dots$ in ${\mathfrak{g}\mathfrak{r}\mathfrak{t}}_1(\mathbb{K})$ with nonzero coefficients of $x^{2k}y$ in ${\sigma }_{2k+1}$ generates a free Lie subalgebra, building on the work of Brown.