One parameter subgroups of a Lie subgroup

Question

Suppose $H$ is a closed connected Lie subgroup of $G$. Are one parameter subgroups of $H$ also one parameter subgroups of $G$? If not true in general, when is the statement true?

Answer 1

Yes, a $1$-parameter subgroup of $G$ is a Lie group homomorphism $\gamma:\mathbb{R}\to G$. If $H$ is a Lie subgroup of $G$ and $\gamma:\mathbb{R}\to H$ a $1$-parameter subgroup of $H$, then the composition $$i\circ\gamma:\mathbb{R}\to H\overset{i}{\hookrightarrow} G$$ is a $1$-parameter subgroup of $G$ since the inclusion $i:H\hookrightarrow G$ is a Lie group homomorphism and a composition of Lie group homomorphisms is a Lie group homomorphism.