If the image of a Lie group homomorphism is included in a subgroup; is it a subgroup of that subgroup?

Question

Let $H$ be a Lie subgroup of $G$. Let $\phi: K \to G$ be a Lie group homormophism with $\phi(K) \subseteq H$. Is $\phi(K)$ a Lie subgroup of $H$?

It is obviously an abstract subgroup, and it remains to check smoothness. Is it always true?

Answer 1

It is not always a subgroup. Consider $T^2$ the two dimensional torus. It is the quotient of $R^2$ by the translations $t_u,t_v$ of respective direction $u$ and $v$. Let $p:R^2\rightarrow T^2$ the covering map. Consider the map $f:R\rightarrow R^2$ defined by $f(x)=x(u+\pi v)$, $p\circ f:R\rightarrow T^2$ is a morphism of Lie groups whose image is not a Lie subgroup of $T^2$ since it is not closed. Here we $G=H=T^2$, $K=R$ and $\phi=p\circ f$.

If $\phi(K)$ is closed in $H$ it is a Lie subgroup of $H$ since a closed subgroup of a Lie group is a Lie subgroup.