Corollary 3

Let G be a finite-dimensional Lie group and $(H_i)_{i \in I}$ a family of Lie subgroups of G. Then $H = \bigcap_{i \in I} H_i$ is a Lie subgroup of G and

There exists a finite subset J of I such that $\bigcap_{i \in J} L(H_i)$ is equal to the intersection M of all the L(H_i). We know that H^* = \bigcap_{i \in J} H_i is a Lie subgroup such that L(H^*) = M (§ 3, no. 8, Corollary 2 to Proposition 29). Let H_0 be the identity component of H^*. It is a Lie subgroup of G and L(H_0) = M. By Corollary 2, H_0 \subset H_i for all i and hence H_0 \subset H \subset H^*, whence the corollary.