Let $G$ be a Lie group and $H$ a Lie subgroup of $G$, i.e. a subgroup in the group theoretic sense and an immersive submanifold. Let $\mathfrak{g}$ and $\mathfrak{h}$ be the associated Lie algebras.
Now, the Lie algebra $\mathfrak{h}$ is given by:
$ \mathfrak{h} = \{ X\in \mathfrak{g} : \exp_G(tX)\in H, \text{ for } \vert t \vert < \varepsilon \text{ for one } \varepsilon > 0 \} $
Thus, this definition varies from the usual one,
$ \mathfrak{h} = \{ X\in \mathfrak{g} : \exp_G(tX)\in H \quad \forall t\in \mathbb{R}\} $
by restricting the values for $t$ to a small interval.
The only thing that we know, is that $H$ is a Lie subgroup of $G$, but how does this property allow for the restriction of the $t$ values?