### An issue with Root systems in sage

Here is a problem that I cannot seem to understand.

```
R=RootSystem(['A', 2])
F=R.ambient_space().fundamental_weights()
D=R.ambient_space().simple_roots()
```

The out-put we get is as follows:

```
F=Finite family {1: (1, 0, 0), 2: (1, 1, 0)}
D = Finite family {1: (1, -1, 0), 2: (0, 1, -1)}
```

` `

The issue is the root lattice is contained in the weight lattice but clearly D[2] is not contained in the lattice generated by F. Perhaps I am missing something simple.

I need to write a program I would need the weyl group action on some weights and roots at the same time. How can this be done?
Thank you for your time in advance.