Whenever possible, avoid working with Sage's symbolic ring.

Here, you can use the symbolic ring as an intermediate step but move to QQbar.

This will make your code work. So you could start as you did:

```
sage: eta = I
sage: eta2 = (1+I)*sqrt(2)/2
```

then instead of defining

```
sage: T = matrix(SR, 4, [eta**(i*j)*eta2/2 for i in range(4) for j in range(4)])
```

you could define

```
sage: T = matrix(QQbar, 4, [eta**(i*j)*eta2/2 for i in range(4) for j in range(4)])
```

And then things work.

```
sage: G = MatrixGroup(T)
sage: G
Matrix group over Algebraic Field with 1 generators (
[ 0.3535533905932738? + 0.3535533905932738?*I 0.3535533905932738? + 0.3535533905932738?*I 0.3535533905932738? + 0.3535533905932738?*I 0.3535533905932738? + 0.3535533905932738?*I]
[ 0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I 0.3535533905932738? - 0.3535533905932738?*I]
[ 0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I 0.3535533905932738? + 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I]
[ 0.3535533905932738? + 0.3535533905932738?*I 0.3535533905932738? - 0.3535533905932738?*I -0.3535533905932738? - 0.3535533905932738?*I -0.3535533905932738? + 0.3535533905932738?*I]
)
```

Calculations in QQbar are exact and efficient.

If you need to see the radical expression of an element in QQbar (if it has one), use `.radical_expression()`

.

For example:

```
sage: T[0,0].radical_expression()
1/2*(-1)^(1/4)
```

I've tried instead:

running into

`NotImplementedError: Currently, only simple algebraic extensions are implemented in gap`

. Well... Perhaps gap cannot digest the input in both cases.Which is the "real need"? Depending on it, the "other way to get the result" may be simpler or more complicated. In the above case one can use block matrices representing $\sqrt{-1}$ and $\sqrt 2$ as commuting $2\times 2$ matrices. If such cases are enough, i will write the code.

Thank you for the comment and time. I have never considered it as block matrices yet. It is very nice idea. I will also try.