Proposition 10

Let H be a Lie subgroup of G and $i : H \to G$ the canonical injection. Then $i_*$ is an injective homomorphism of the algebra $\mathscr{T}^{(\infty)}(H)$ into the algebra $\mathscr{T}^{(\infty)}(H)$ and $i_*(t^\vee) = (i_*(t))^\vee$ for all $t \in \mathscr{T}^{(\infty)}(H)$.

This follows from Propositions 6 and 8 and Differentiable and Analytic Manifolds, R, 13.2.3.

$\mathscr{T}^{(\infty)}(H)$ is identified with a subalgebra of $\mathscr{T}^{(\infty)}(G)$ by means of the isomorphism of Proposition 10.