Definition 7

Let X be a manifold of class $C^r$ ($r \geq 2$) and $g$ a complete normable Lie algebra. A law of infinitesimal left (resp. right) operation of class $C^{r-1}$ of $g$ on X is a mapping $a \mapsto D_a$ of $g$ into the set of vector fields on X with the following properties:

(a) the mapping $(a, x) \mapsto D_a(x)$ is a morphism of class $C^{r-1}$ of the trivial vector bundle $g \times X$ into the vector bundle $T(X)$;

(b) $[D_a, D_b] = -D_{[a, b]}$ (resp. $[D_a, D_b] = D_{[a, b]}$ for all $a, b$ in $g$).

In particular, each vector field $D_a$ is of class $C^{r-1}$.