This is probably trivial (hope not!), but I am looking for a proof that if G is a Lie subgroup of H and G is also the connected component of H, then G and H have isomorphic Lie algebras. If this does not hold, what other restrictions on G and H could make it work?
Thank you!
The Lie algebra "is" the tangent space at the identity, i.e. the derivatives at $t = 0$ of smooth curves $c(t)$ with $c(0) = e$. These curves necessarily have range in the connected component, so you get the same result whether you consider the whole group or just the connected component.