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On $(A_2,A_6)$
sol 1
$$ \begin{bmatrix}
z_0^{(1)} & z_0^{(2)} \\
z_1^{(1)} & z_1^{(2)} \\
z_2^{(1)} & z_2^{(2)} \\
z_3^{(1)} & z_3^{(2)} \\
z_4^{(1)} & z_4^{(2)} \\
z_5^{(1)} & z_5^{(2)} \\
z_6^{(1)} & z_6^{(2)} \\
z_7^{(1)} & z_7^{(2)}
\end{bmatrix} = \begin{bmatrix}
1. & 1. \\ 2.24698 & 2.24698 \\ 2.80194 & 2.80194 \\ 2.24698 & 2.24698 \\ 1. & 1. \\ 0 & 0 \\ 0 & 0 \\ 1. & 1.
\end{bmatrix} $$ Here $2.24698\cdots$ is a solution of $x^3-2 x^2-x+1=0$.
This solution can be obtained from $g\in SU(3)$ given by $$ g=\left( \begin{array}{ccc}
e^{\frac{2 i \pi }{7}} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^{-\frac{2 i \pi }{7}} \\
\end{array} \right) $$
sol 2
$$ \begin{bmatrix}
z_0^{(1)} & z_0^{(2)} \\
z_1^{(1)} & z_1^{(2)} \\
z_2^{(1)} & z_2^{(2)} \\
z_3^{(1)} & z_3^{(2)} \\
z_4^{(1)} & z_4^{(2)} \\
z_5^{(1)} & z_5^{(2)} \\
z_6^{(1)} & z_6^{(2)} \\
z_7^{(1)} & z_7^{(2)}
\end{bmatrix} = \begin{bmatrix}
1. & 1. \\ 1.61803 & 1.61803 \\ 1. & 1. \\ 0 & 0 \\ 0 & 0 \\ 1. & 1. \\ 1.61803 & 1.61803 \\ 1. & 1.
\end{bmatrix} $$ where $1.61803\cdots$ is the Golden ratio.
This solution can be obtained from $g\in SU(3)$ given by $$ g=\left( \begin{array}{ccc}
e^{\frac{2 i \pi }{5}} & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & e^{-\frac{2 i \pi }{5}} \\
\end{array} \right) $$