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.