 Elementary unipotent generators of SL2R

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.