Lemma 1

Let G be a Lie group and $\alpha$ a vector field on G. For all $g \in G$, let

The following conditions are equivalent: (i) $\alpha$ is a homomorphism of the group G into the group T(G); (ii) for all $g, g'$ in G, $\alpha(gg') = \alpha(g)g' + g\alpha(g')$; (iii) for all $g, g'$ in G, $\beta(gg_1) = \beta(g) + (\mathrm{Ad}\,g)\beta(g')$.

Condition (i) means that, for all $g, g'$ in G, we have in the group T(G):

or But the product of $\beta(g)$ and $(\mathrm{Ad}\,g)\beta(g')$ in T(G) is just the sum of $\beta(g)$ and $(\mathrm{Ad}\,g)\beta(g')$ in $L(G)$ (§ 2, no. 1, Proposition 2). Hence (i) $\Leftrightarrow$ (iii). On the other hand, condition (ii) may be written $\beta(gg')gg' = \beta(g)gg' + g\beta(g')g'$, or and hence (ii) $\Leftrightarrow$ (iii).