Proposition 5

Let G be a connected real Lie group. Suppose that $L(G)$ has a complex normable Lie algebra structure $L'$ compatible with its real normable Lie algebra structure. There exists on G one and only one complex Lie group structure compatible with the real Lie group structure and with Lie algebra $L'$.

By $§ 4$, no. 2, Corollary 2 to Theorem 2, it suffices to prove that the structure of $L'$ is invariant under Ad G. Let $\phi$ be an exponential mapping of G. By $§ 4$, no. 4, Corollary 3 (i) to Proposition 8, there exists a neighbourhood V of 0 in $L(G)$ such that the structure of $L'$ is invariant under Ad $\phi(V)$. But Ad $\phi(V)$ generates G because G is connected.

The conclusion of Proposition 5 is not necessarily true if G is not assumed to be connected (Exercise 7).