Proposition 30

Let G be a finite-dimensional Lie group with a countable base and

H and H' Lie subgroups of G. Suppose that K is of characteristic 0 and that HH' is locally closed in G.

(i) HH' is a submanifold of G and $T_e(HH') = L(H) + L(H')$.

(ii) Suppose that every element of H commutes with every element of H'. Then HH' is a Lie subgroup of G. Let $\phi$ be the mapping $(h, h') \mapsto hh'$ of $H \times H'$ onto HH'. The kernel of $\phi$ is the set of $(m, m^{-1})$ where $m \in H \cap H'$ and the morphism of $(H \times H')/\mathrm{Ker}\ \phi$ onto HH' derived from $\phi$ by passing to the quotient is a Lie group isomorphism.

Let $H \times H'$ operate on G on the right by the mapping $((h, h'), g) \mapsto hg{h'}^{-1}$. The orbital mapping $\rho$ of e is $(h, h') \mapsto h{h'}^{-1}$. By Proposition 14 (iii) of § 1, no. 7, HH' is a submanifold of G and $T_e(HH') = \mathrm{Im}\ T_e(\rho)$. Now

and hence $T_e(HH') = L(H) + L(H')$. Suppose that every element of H commutes with every element of H'. Then HH' is a subgroup of G. By (i), it is a Lie subgroup of G. The rest of the statement follows from Proposition 28.