Remark

Let X be a manifold of class $C^r$, $g$ a finite-dimensional Lie algebra and $a \mapsto D_a$ a linear mapping of $g$ into the vector space of vector fields of class $C^{r-1}$ on X. Then condition (a) of Definition 7 holds. For, by considering a basis of $g$ and applying Differentiable and Analytic Manifolds, R, 7.7.1, the problem is reduced to the case $\dim g = 1$ and our assertion is then obvious.