Real special linear group of degree two ($S{L}_2(\mathbb{R})$)

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.

The upper and lower unipotent matrices generate SL2R. For $a\ne 0$, let $w(a)=U(a)L(-a^{-1})U(a)$; multiplication gives $w(a)=\left(\begin{array}{cc}0& a\\ -a^{-1}& 0\end{array}\right)$ and $w(a)w(-1)=\mathrm{diag}(a,a^{-1})$. A determinant-one matrix with upper-left entry $a\ne 0$ factors as $L(c/a)\mathrm{diag}(a,a^{-1})U(b/a)$. If that entry vanishes, first add the second row to the first by $U(1)$. Lower unipotents are conjugates of upper unipotents, so the latter also generate the whole group normally.