Definition 3

The Lie group structure on $H$ defined in Proposition 9 is called the structure induced on $H$ by the Lie group structure on $G$.

If $H$ is a Lie subgroup of $G$, its Lie group structure is induced by that on $G$ (Differentiable and Analytic Manifolds, R, 5.8.5).

If $G = \mathbf{R}$ and $H = \mathbf{Q}$, then $\mathfrak{h} = \{0\}$ and hence the structure induced on $H$ is the discrete Lie group structure. Similarly if $G = \mathbf{C}$ (considered as a complex Lie group) and $H = \mathbf{R}$.