These two matrices generate a free group: $$ \left( \begin{array}{ccc} \frac{1}{3} & \frac{2 \sqrt{2}}{3} & 0 \\ -\frac{2 \sqrt{2}}{3} & \frac{1}{3} & 0 \\ 0 & 0 & 1 \end{array} \right), \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & \frac{1}{3} & \frac{2 \sqrt{2}}{3} \\ 0 & -\frac{\sqrt{2}}{3} & \frac{1}{3} \end{array} \right). $$ To see that they do, consider a set of corresponding matrices over the field $\mathbb{F}_3(i)$:
$$ \left( \begin{array}{ccc} 1 & \mp i & 0 \\ \pm i & 1 & 0 \\ 0 & 0 & 0 \end{array} \right), \left( \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 1 & \mp i \\ 0 & \pm i & 1 \end{array} \right). $$ Non-trivial reduced words in these matrices always have disagreement along the diagonal.