  Two doubly indexed families of homogeneous and isobaric polynomials in
several indeterminates are considered: the (partial) exponential Bell
polynomials $B_{n,k}$ and a new family $S_{n,k}$ such that
$X_1^{-(2n-1)}S_{n,k}$ and $B_{n,k}$ obey an inversion law which generalizes
that of the Stirling numbers of the first and second kind. Both polynomial
families appear as Lie coefficients in expansions of certain derivatives of
higher order. Substituting $D^j(\varphi)$ (the $j$-th derivative of a fixed
function $\varphi$) in place of the indeterminates $X_j$ shows that both
$S_{n,k}$ and $B_{n,k}$ are differential polynomials depending on $\varphi$ and
on its inverse $\overline{\varphi}$, respectively. Some new light is shed
thereby on Comtet's solution of the Lagrange inversion problem in terms of the
Bell polynomials. According to Haiman and Schmitt that solution is essentially
the antipode on the Fa\`{a} di Bruno Hopf algebra. It can be represented by
$X_1^{-(2n-1)}S_{n,1}$. Moreover, a general expansion formula that holds for
the whole family $S_{n,k}$ ($1\leq k\leq n$) is established together with a
closed expression for the coefficients of $S_{n,k}$. Several important
properties of the Stirling numbers are demonstrated to be special cases of
relations between the corresponding polynomials. As a non-trivial example, a
Schl\"omilch-type formula is derived expressing $S_{n,k}$ in terms of the Bell
polynomials $B_{n,k}$, and vice versa.
