Branching polytopes for classical Lie algebras over $A_{n-1}$

We describe the branching of Lie algebras of classical type over $A_{n-1}$ using an inductive approach, which was motivated by the work of Gornitskii. This allows us to label the highest weight vectors of the modules occurring in the decomposition of the restriction of a finite-dimensional simple module to $A_{n-1}$ by lattice points of a string or a Lusztig polytope.