I have a proposition I would like to prove that says that if I have a Lie group morphism $f:H \mapsto G$ that is also an injective immersion, then $h \subset g$ is a Lie subalgebra. Now, there is a result stating that if I have a Lie group morphism $f$, then $f_{*}[v,w]= [f_{*}v,f_{*}v]$, so $f_{*} (h)$ is a Lie subalgebra. I also understand why you require the morphism to be an immersion, since this way $f_{*}= df_{e}:h \mapsto g$ would be injective, and the image of $h$ through $f_{*}$ on $g$ would be isomorphic to $h$. So I have $h \cong f_{*}(h) \subset g$. I wonder, though, why do you require $f$ to be injective?
We have defined Lie subgroups as images of injective immersion, because this way there is a powerful theorem that gives us a bijection between Lie subgroups and Lie subalgebras. I just don't get why you ask for injectivity.