Let $G$ and $H$ be two Lie groups with the same Lie algebra $\mathfrak g=\mathfrak h$ and let $f:G\to H$ be a Lie group homomorphism that is also a local diffeomorphism. I claim that the differential map at the identity $d_ef:T_eG\to T_eH$ is the identity map of $\mathfrak g$, however, I am not able to prove it.
Is my claim true?
Thanks in advance.