If a shuffle-finite series has Lie rank $k$, its minimal formal realization can be chosen with shuffle-finite coordinate series. They can lie in any finitely generated, derivative-stable shuffle algebra containing the series.

For a complement $P_j$ to the residual stabilizer, choose polynomials $Q_i$ with $(f,Q_iP_j)=\delta_{ij}$. The coordinate series are $g_i=f\circ Q_i-(f,Q_i)1$. They belong to the witnessing algebra and give formal coordinates on the stabilizer ideal's annihilator. This does not assert polynomial output or transition functions in these coordinates.