Proposition 29

Let G and H be Lie groups and φ a morphism of G into H. Suppose that K is of characteristic 0 and that H is finite-dimensional. If H' is a Lie subgroup of H, then G' = φ⁻¹(H') is a Lie subgroup of G and L(G') = L(φ)⁻¹(L(H')).

Let π be the canonical mapping of H into the homogeneous space X = H/H'. Let G operate on X on the left by the mapping (g, x) ↦ φ(g)x. The stabilizer of π(e) is G', which is therefore a Lie subgroup of G (§ 1, no. 7, Proposition 14). The orbital mapping of π(e) is π ∘ φ. By Proposition 14 of § 1, L(G') is the kernel of L(π ∘ φ) = T_e(π) ∘ L(φ). The kernel of T_e(π) is L(H') (§ 1, no. 6, Proposition 11 (i)) and hence Ker L(π ∘ φ) = L(φ)⁻¹(L(H')).