I'm given to understand the exponential map is not generally surjective -- the standard example is $\mathrm{SL}(\mathbb{R}^2)$ [ 1 ].
I can clearly see why this is so in the non-connected case -- the tangent space is a tangent space to the connected component alone, so its image must be contained in the connected component. I do not see why the map isn't surjective in the connected case.
I also don't see why the map is then again surjective in the compact case -- wikipedia claims that this is a special case of "the exponential map is surjective if every element is contained in a maximal torus". Is this right? Is there a good way to understand why this is true?
Note that I am not looking for counter-examples: I'm aware of them. I'm looking for intuition -- perhaps a clever look at what the image of the exponential map actually looks like in the non-surjective case (how it "misses" some of the points in the group).
As an analogy, if asked to explain smooth non-analytic functions, it would be more instructive (than simply providing the example of $e^{-1/x}$) to explain that a function may grow slower than all polynomials near zero -- and provide the construction as $1/f(1/x)$ from any function $f$ that grows faster than all polynomials as $x\to\infty$.
(See here for more examples of the kind of intuition I'm looking for, within the context of Lie theory.)
