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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?

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up vote 7 down vote accepted

If $X\in\mathfrak g$ is such that $\exp(tX)\in H$ for small values of $t$, then in fact $\exp(tX)$ is in $H$ for all values of $t$.

Indeed, suppose you know that $\exp(tX)\in H$ for $t\in(-t_0,t_0)$ and let $t\in\mathbb R$. There exists $n\in\mathbb N$ such that $t/n\in(-t_0,t_0)$, and then $$\exp(tX)=\exp(n\cdot\tfrac tnX)=\exp(\tfrac tnX)^n,$$ and the latter is in $H$ because $\exp(\tfrac tnX)$ is and $H$ is a subgroup.

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