Proposition 6

Let $G$ be a Lie group. (i) The canonical projection $p : T(G) \to G$ is a Lie group morphism. (ii) The kernel of $p$ is $T_e(G)$. It is a Lie subgroup of $T(G)$. The Lie group structure induced on $T_e(G)$ by that on $T(G)$ is the Lie group structure of the complete normable space $T_e(G)$. (iii) The zero section $s$ is an isomorphism of the Lie group $G$ onto a Lie subgroup $s(G)$ of $T(G)$ (which subgroup we identify with $G$). (iv) The Lie group $T(G)$ is the semi-direct product of $G$ by $T_e(G)$.

Assertion (i) follows from (5). Assertion (ii) is obvious taking account of Proposition 2 (ii). Assertions (iii) and (iv) follow from (6) and § 1, Proposition 8.

Let $u \in T(G)$ and $g \in G$. By (3) and (4), the products $ug, gu$ calculated in the group $T(G)$ are the images of $u$ under $T(\delta(g^{-1}))$ and $T(\gamma(g))$. It follows from § 1, Corollary 2 to Proposition 17 that the mapping $(g, u) \mapsto gu$ of $G \times T_e(G)$ into $T(G)$ is an isomorphism of the trivial vector bundle $G \times T_e(G)$ with base space $G$ onto the vector bundle $T(G)$. The inverse isomorphism is called the left trivialization of $T(G)$. By considering the mapping $(g, u) \mapsto ug$, the right trivialization of $T(G)$ is defined similarly.