Theorem 1

Let G and H be Lie group germs and f a continuous morphism of $L(G)$ into $L(H)$. (i) There exist an open Lie subgroup germ $G'$ of G and a morphism $\phi$ of $G'$ into H such that $f = L(\phi)$. (ii) Let $G_1, G_2$ be open Lie subgroup germs of G and $\phi_i$ a morphism of $G_i$ into H such that $f = L(\phi_i)$ for $i = 1, 2$. Then $\phi_1$ and $\phi_2$ coincide on a neighbourhood of e. Let $p_1 : G \times H \to G, p_2 : G \times H \to H$ be the canonical projections. For all $(g, h) \in G \times H$, let $f_{g, h}$ be the mapping $ga \mapsto hf(a)$ of $T_g(G) = gL(G)$ into $T_h(H) = hL(H)$. By considering the left trivializations of $T(G)$ and $T(H)$, it is seen immediately that the $f_{g, h}$ define a morphism of $p_1^*T(G)$ into $p_2^*T(H)$. Let $a$ be the graph of $f$; it is a closed Lie subalgebra of $L(G) \times L(H)$ which admits $\{0\} \times L(H)$ as topological supplement. For all $(g, h) \in G \times H$, the graph of $f_{g,h}$ is $(g, h) \cdot a$. The union of these graphs is an integrable vector subbundle of $T(G \times H)$ (Lemma 1). Then there exist (Differentiable and Analytic Manifolds, R, 9.3.7) an open neighbourhood $U$ of $e_G$ in $G$ and an analytic mapping $\phi$ of $U$ into $H$ such that $\phi(e_G) = e_H$ and $T_g(\phi) = f_{g, \phi(g)}$ for all $g \in U$. In particular, $T_{e_G}(\phi) = f$.

Let $V$ be an open neighbourhood of $e_G$ in $G$ such that, for $(s, t) \in V \times V$, the products $st$ and $\phi(s)\phi(t)$ are defined and $st \in U$. Consider the mappings $\alpha_1, \alpha_2$ of $V \times V$ into $H$ defined by

Then $\alpha_1(t, e) = \phi(t) = \alpha_2(t, e)$. On the other hand, let $t$ be fixed in $V$ and $\beta_i$ be the mapping $s \mapsto \alpha_i(s, t)$ of $V$ into $H$. Then, for all $s \in V$ and all $a \in L(G)$,

Hence (Differentiable and Analytic Manifolds, R, 9.3.7) $\alpha_1$ and $\alpha_2$ coincide on a neighbourhood of $(e_G, e_G)$. The restriction of $\phi$ to a sufficiently small symmetric open neighbourhood of $e_G$ is therefore a morphism of Lie group germs, whence (i).

Let $G_1, G_2, \phi_1, \phi_2$ be as in (ii) and let us prove that $\phi_1, \phi_2$ coincide on a neighbourhood of $e_G$. There exists an open neighbourhood $W$ of $e_G$ such that $\phi_1(ts) = \phi_1(t)\phi_1(s)$, $\phi_2(ts) = \phi_2(t)\phi_2(s)$ for all $s, t$ in $W$. Then, if $s \in W$ and $a \in L(G)$,

for $i = 1, 2$. As $\phi_1(e_G) = e_H = \phi_2(e_G)$, it follows from Differentiable and Analytic Manifolds, R, 9.3.7 that $\phi_1$ and $\phi_2$ coincide on a neighbourhood of $e_G$.