Non-surjectivity of $\exp$ map for non-compact Lie group.

Question

I have often read that the exponential map from the Lie algebra of a non-compact Lie group is not surjective, however the product of exponentials involving the compact and non-compact generators of the algebra is surjective. At the moment such a proof is beyond my capability, might someone be able to give me a reference to this result (preferably the original paper, I think it may have been due to Cartan)?

Answer 1

For some non-compact Lie groups such as $\text{SL}_2(\Bbb R)$ it is the case that not every group element is a power of a Lie group element. See Proving that any element of $\text{SL}(2,\mathbb{R})$ can be expressed as $\pm\exp(z)$.

On the other hand, there are non-compact groups such as $\Bbb R^n$ (under addition) where every group element is a power of a Lie group element.