Given a semisimple Lie algebra L with a split Cartan subalgebra, this function computes the root system. This function returns four values:

• A sequence R giving the set of roots of L with respect to the Cartan subalgebra which is output by CartanSubalgebra(L). The first half of the sequence consists of the positive roots, listed according to increasing height and the second half of the sequence consists of the negative roots.

• A sequence of elements of L wich are the root vectors corresponding to the roots in R (so the first vector corresponds to the first root and so on).

• A sequence of fundamental roots.

• The Cartan matrix of the sequence of fundamental roots.

> L := SimpleLieAlgebra("G", 2, RationalField());
> R, Rv, fund, C:=RootSystem(L);
> R;
[
    ( 1 -2),
    (0 1),
    ( 1 -1),
    (1 0),
    (1 1),
    ( 2 -1),
    (-1  2),
    ( 0 -1),
    (-1  1),
    (-1  0),
    (-1 -1),
    (-2  1)
]
> Rv;
[ (1 0 0 0 0 0 0 0 0 0 0 0 0 0), (0 1 0 0 0 0 0 0 0 0 0 0 0 0), 
(0 0 1 0 0 0 0 0 0 0 0 0 0 0), (0 0 0 0 1 0 0 0 0 0 0 0 0 0),
(0 0 0 1 0 0 0 0 0 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 1 0 0 0 0),
(0 0 0 0 0 0 0 0 0 0 1 0 0 0), (0 0 0 0 0 0 1 0 0 0 0 0 0 0),
(0 0 0 0 0 0 0 0 1 0 0 0 0 0), (0 0 0 0 0 0 0 0 0 0 0 1 0 0),
(0 0 0 0 0 0 0 0 0 0 0 0 0 1), (0 0 0 0 0 0 0 0 0 0 0 0 1 0) ]
> fund;
[
    ( 1 -2),
    (0 1)
]
> C;
[
    [ 2, -3 ],
    [ -1, 2 ]
]