The Chen-Fliess series

The P. Hall basis from Section 15.4.3 applies in general to any Lie algebra. Let $B_1$, $\ldots$, $B_s$ denote a P. Hall basis for the nilpotent formal Lie algebra $L_k(y_1,\ldots,y_m)$. An important theorem in the study of formal Lie groups is that every $S \in G_k(y_1,\ldots,y_m)$ can be expressed in terms of the P. Hall basis of its formal Lie algebra as

$$S = e^{z_s B_s} e^{z_{s-1} B_{s-1}} \cdots e^{z_2 B_2} e^{z_1 B_1} ,$$ (15.127)

which is called the Chen-Fliess series. The $z_i$ are sometimes called the backward P. Hall coordinates of $S$ (there is a forward version, for which the terms in (15.127) go from $1$ to $s$, instead of $s$ to $1$).