= Elementary unipotent generators of SL2R
{title2=$U(s)=\begin{pmatrix}1&s\\0&1\end{pmatrix},\quad L(t)=\begin{pmatrix}1&0\\t&1\end{pmatrix}$}
The upper and lower <unipotent matrices> generate <SL2R>. For $a\ne0$, let $w(a)=U(a)L(-a^{-1})U(a)$; multiplication gives $w(a)=\begin{pmatrix}0&a\\-a^{-1}&0\end{pmatrix}$ and $w(a)w(-1)=\operatorname{diag}(a,a^{-1})$. A determinant-one matrix with upper-left entry $a\ne0$ factors as $L(c/a)\operatorname{diag}(a,a^{-1})U(b/a)$. 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.
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