Corollary

Let $n \in \mathbf{Z}$. The tangent mapping at $e$ to the mapping $g \mapsto g^n$ of $G$ into $G$ is the mapping $x \mapsto nx$ of $T_e(G)$ into $T_e(G)$.

For $n \geqslant 0$, this follows by induction on $n$ from Proposition 7. On the other hand, the tangent mapping at $e$ to the mapping $g \mapsto g^{-1}$ is the mapping $x \mapsto -x$ (no. 1, Proposition 2).

Let G be a Lie group, X a manifold of class C^r and (g, x) \mapsto gx a law of left operation of class C^r of G on X. Arguing as for Proposition 1, we derive a law of left operation of class C^{r-1} of T(G) on T(X), which we shall also denote by (u, v) \mapsto uv. Identifying G (resp. X) with the image of the zero section of T(G) (resp. T(X)), we see by (6) that the law of left operation of T(G) on T(X) extends the law of left operation of G on X. For all u \in T_g(G) and v \in T_x(X), by (1),

If g \in G and v \in T_x(X), gv is, by (3), the image of v under the tangent mapping at x to the mapping y \mapsto gy of X into X. This tangent mapping is an isomorphism of T_x(X) onto T_{gx}(X). In particular,

If x \in X and u \in T_g(G), ux is by (4) the image of u under the tangent mapping at g to the mapping h \mapsto hx of G into X. Hence

The above can be applied to the case of a Lie group operating on itself by left (resp. right) translation. The corresponding law of operation of T(G) on T(G) is defined by left (resp. right) translation of the Lie group T(G). Formulae (7), (8) and (9) are therefore valid in T(G).