Proposition 3

Let G be a Lie group germ, L its Lie algebra and $\phi$ an exponential mapping of G. Let $L_1, \ldots, L_n$ be vector subspaces of L such that L is the topological direct sum of $L_1, \ldots, L_n$. The mapping

defined on an open subset of $L_1 \times L_2 \times \cdots \times L_n$, is analytic. The tangent mapping at $(0, 0, \ldots, 0)$ to $\theta$ is the canonical mapping of $L_1 \times \cdots \times L_n$ into L.

Let $k_i$ be the canonical injection of $L_i$ into $L_1 \times L_2 \times \cdots \times L_n$. Then, for all $b \in L_i$, $(T_{(0, \ldots, 0)}\theta)(T_0k_i)(b) = (T_0\phi)(b) = b$ and hence $(T_{(0, \ldots, 0)}\theta)|L_i$ is the canonical injection of $L_i$ into L.

In particular, $\theta$ is étale at $(0, 0, \ldots, 0)$. Its restriction to a sufficiently small

LIE GROUPS

open neighbourhood U of $(0, 0, \ldots, 0)$ has an open image in G and is an isomorphism of the manifold U onto the manifold $\theta(U)$. The inverse mapping $\eta$ of $\theta(U)$ onto U is called a canonical chart of the second species of G, associated with the given decomposition of L as a direct sum. If further G is finite-dimensional and each $L_i$ is generated by a non-zero vector $e_i$, the coordinate system on $\theta(U)$ defined by $\eta$ and the $e_i$ is called a canonical coordinate system of the second species.