A false belief which I meet not infrequently this time of year while marking exams is the following:
The exponential map is surjective for a connected Lie group.
This is true for compact Lie groups, but certainly false in general. A (finite-dimensional) connected Lie group is generated by the image of the exponential map, but already $SL(2,\mathbb{R})$ shows that there are elements which are not in the image of the exponential map.
Interestingly, for a connected real Lie group, every element can be written as the product of at most two exponentials.
