Lemma 1

Let G be a Lie group germ and $\mathfrak{h}$ a Lie subalgebra of $L(G)$ admitting a topological supplement. The union of the $g\mathfrak{h}$ (resp. $\mathfrak{h}g$) for $g \in G$ is an integrable vector subbundle of $T(G)$.

By considering the left trivialization of $T(G)$ (§ 2, no. 3), it is seen immediately that the $g\mathfrak{h}$, for $g \in G$, are the fibres of a vector subbundle E of $T(G)$. Let $g \in G$. The set of $(L_a)_g$, where $a \in \mathfrak{h}$, is equal to $g\mathfrak{h}$. Now, if $a$ and $b$ belong to $\mathfrak{h}$, then $[L_a, L_b] = L_{[a, b]}$ and $[a, b] \in \mathfrak{h}$. Hence E is integrable (Differentiable and Analytic Manifolds, R, 9.3.3 (iv)). The argument is similar for the $\mathfrak{h}g$.

The integral foliation (Differentiable and Analytic Manifolds, R, 9.3.2) of the union of the $g\mathfrak{h}$ (resp. $\mathfrak{h}g$) is called the left (resp. right) foliation of G associated with $\mathfrak{h}$.