In the SAGE computer package, there useful exist tools for branching representations of a simple Lie group to a Levi subgroup:
Explicitly, one is branching to subgroup corresponding to a Dynkin sub-diagram, obtained by removing a single node.
For example, we can branch from $SL(n)$ to the subgroup $SL(n-1)$. However, $SL(n-1)$ can be considered as "living" in the larger subgroup $SL(n-1) \times U1$. This is true for every subgroup coming from a deleted node, i.e. one can always take the product of the subgroup with U1, to obtain a larger subgroup.
How does one branch to this subgroup in SAGE. For example, it is done in the LieArt program for mathematica: see A3 of the following
Is this also possible in SAGE?