 Real special linear group of degree two

This Lie group consists of real two-by-two matrices of determinant one. Its Lie algebra is the space of traceless real two-by-two matrices. Its Exponential map of a Lie group is not onto: by the Cayley-Hamilton theorem, $A^2=-\det (A)I$ for traceless $A$, and its exponential has trace either $2\cosh s$ or $2\cos s$, hence never less than $-2$. The determinant-one matrix $\mathrm{diag}(-2,-1/2)$ is therefore outside its exponential image.