Theorem 2
Tag 0AH4, permanent. Cite it as [Bourbaki, Tag 0AH4].
Lie Groups and Lie Algebras, chapter III, § 7. Lie groups over an ultrametric field
Let G be a Lie group and h a Lie subalgebra of L(G) admitting a topological supplement. There exists a Lie subgroup H of G such that L(H) = h. If H_1 and H_2 are Lie subgroups of G such that L(H_1) = L(H_2) = h, then H_1 \cap H_2 is open in H_1 and H_2.
The first assertion follows from Proposition 1 and § 4, no. 2, Theorem 3. The second is a special case of § 4, no. 2, Theorem 3.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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